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Theorem kqnrmlem2 23641

Description: If the Kolmogorov quotient of a space is normal then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)

**Hypothesis**
Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

**Assertion**
Ref	Expression
kqnrmlem2	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)

Distinct variable groups: 𝑥,𝑦,𝐽 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦)

Proof of Theorem kqnrmlem2 Dummy variables 𝑚 𝑤 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 22808	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2	1	adantr 480	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Top)
3		simplr 768	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (KQ‘𝐽) ∈ Nrm)
4		simpll 766	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝐽 ∈ (TopOn‘𝑋))
5		simprl 770	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑧 ∈ 𝐽)
6		kqval.2	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
7	6	kqopn 23631	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
8	4, 5, 7	syl2anc 583	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽))
9		simprr 772	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))
10	9	elin1d 4194	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘𝐽))
11	6	kqcld 23632	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝑤) ∈ (Clsd‘(KQ‘𝐽)))
12	4, 10, 11	syl2anc 583	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹 “ 𝑤) ∈ (Clsd‘(KQ‘𝐽)))
13	9	elin2d 4195	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
14		elpwi 4605	. . . . . 6 ⊢ (𝑤 ∈ 𝒫 𝑧 → 𝑤 ⊆ 𝑧)
15		imass2 6100	. . . . . 6 ⊢ (𝑤 ⊆ 𝑧 → (𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑧))
16	13, 14, 15	3syl 18	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑧))
17		nrmsep3 23252	. . . . 5 ⊢ (((KQ‘𝐽) ∈ Nrm ∧ ((𝐹 “ 𝑧) ∈ (KQ‘𝐽) ∧ (𝐹 “ 𝑤) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))
18	3, 8, 12, 16, 17	syl13anc 1370	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))
19		simplll 774	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝐽 ∈ (TopOn‘𝑋))
20	6	kqid 23625	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
21	19, 20	syl 17	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
22		simprl 770	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑚 ∈ (KQ‘𝐽))
23		cnima 23162	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑚 ∈ (KQ‘𝐽)) → (^◡𝐹 “ 𝑚) ∈ 𝐽)
24	21, 22, 23	syl2anc 583	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ 𝑚) ∈ 𝐽)
25		simprrl 780	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (𝐹 “ 𝑤) ⊆ 𝑚)
26	6	kqffn 23622	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27		fnfun 6648	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
28	19, 26, 27	3syl 18	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → Fun 𝐹)
29	10	adantr 480	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑤 ∈ (Clsd‘𝐽))
30		eqid 2728	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
31	30	cldss 22926	. . . . . . . . 9 ⊢ (𝑤 ∈ (Clsd‘𝐽) → 𝑤 ⊆ ∪ 𝐽)
32	29, 31	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑤 ⊆ ∪ 𝐽)
33		fndm 6651	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
34	19, 26, 33	3syl 18	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → dom 𝐹 = 𝑋)
35		toponuni 22809	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
36	19, 35	syl 17	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑋 = ∪ 𝐽)
37	34, 36	eqtrd 2768	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → dom 𝐹 = ∪ 𝐽)
38	32, 37	sseqtrrd 4019	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑤 ⊆ dom 𝐹)
39		funimass3 7057	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ dom 𝐹) → ((𝐹 “ 𝑤) ⊆ 𝑚 ↔ 𝑤 ⊆ (^◡𝐹 “ 𝑚)))
40	28, 38, 39	syl2anc 583	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → ((𝐹 “ 𝑤) ⊆ 𝑚 ↔ 𝑤 ⊆ (^◡𝐹 “ 𝑚)))
41	25, 40	mpbid 231	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑤 ⊆ (^◡𝐹 “ 𝑚))
42	6	kqtopon 23624	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
43		topontop 22808	. . . . . . . . . 10 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
44	19, 42, 43	3syl 18	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (KQ‘𝐽) ∈ Top)
45		elssuni 4935	. . . . . . . . . 10 ⊢ (𝑚 ∈ (KQ‘𝐽) → 𝑚 ⊆ ∪ (KQ‘𝐽))
46	45	ad2antrl 727	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑚 ⊆ ∪ (KQ‘𝐽))
47		eqid 2728	. . . . . . . . . 10 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
48	47	clscld 22944	. . . . . . . . 9 ⊢ (((KQ‘𝐽) ∈ Top ∧ 𝑚 ⊆ ∪ (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽)))
49	44, 46, 48	syl2anc 583	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽)))
50		cnclima 23165	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ∈ (Clsd‘𝐽))
51	21, 49, 50	syl2anc 583	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ∈ (Clsd‘𝐽))
52	47	sscls 22953	. . . . . . . . 9 ⊢ (((KQ‘𝐽) ∈ Top ∧ 𝑚 ⊆ ∪ (KQ‘𝐽)) → 𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚))
53	44, 46, 52	syl2anc 583	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚))
54		imass2 6100	. . . . . . . 8 ⊢ (𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚) → (^◡𝐹 “ 𝑚) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
55	53, 54	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ 𝑚) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
56	30	clsss2 22969	. . . . . . 7 ⊢ (((^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ∈ (Clsd‘𝐽) ∧ (^◡𝐹 “ 𝑚) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚))) → ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
57	51, 55, 56	syl2anc 583	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
58		simprrr 781	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧))
59		imass2 6100	. . . . . . . 8 ⊢ (((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ⊆ (^◡𝐹 “ (𝐹 “ 𝑧)))
60	58, 59	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ⊆ (^◡𝐹 “ (𝐹 “ 𝑧)))
61	5	adantr 480	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → 𝑧 ∈ 𝐽)
62	6	kqsat 23628	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (^◡𝐹 “ (𝐹 “ 𝑧)) = 𝑧)
63	19, 61, 62	syl2anc 583	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ (𝐹 “ 𝑧)) = 𝑧)
64	60, 63	sseqtrd 4018	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → (^◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ⊆ 𝑧)
65	57, 64	sstrd 3988	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ 𝑧)
66		sseq2 4004	. . . . . . 7 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → (𝑤 ⊆ 𝑢 ↔ 𝑤 ⊆ (^◡𝐹 “ 𝑚)))
67		fveq2 6891	. . . . . . . 8 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → ((cls‘𝐽)‘𝑢) = ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)))
68	67	sseq1d 4009	. . . . . . 7 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → (((cls‘𝐽)‘𝑢) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ 𝑧))
69	66, 68	anbi12d 631	. . . . . 6 ⊢ (𝑢 = (^◡𝐹 “ 𝑚) → ((𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧) ↔ (𝑤 ⊆ (^◡𝐹 “ 𝑚) ∧ ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ 𝑧)))
70	69	rspcev 3608	. . . . 5 ⊢ (((^◡𝐹 “ 𝑚) ∈ 𝐽 ∧ (𝑤 ⊆ (^◡𝐹 “ 𝑚) ∧ ((cls‘𝐽)‘(^◡𝐹 “ 𝑚)) ⊆ 𝑧)) → ∃𝑢 ∈ 𝐽 (𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
71	24, 41, 65, 70	syl12anc 836	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹 “ 𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹 “ 𝑧)))) → ∃𝑢 ∈ 𝐽 (𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
72	18, 71	rexlimddv 3157	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → ∃𝑢 ∈ 𝐽 (𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
73	72	ralrimivva 3196	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → ∀𝑧 ∈ 𝐽 ∀𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧)∃𝑢 ∈ 𝐽 (𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
74		isnrm 23232	. 2 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑧 ∈ 𝐽 ∀𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧)∃𝑢 ∈ 𝐽 (𝑤 ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧)))
75	2, 73, 74	sylanbrc 582	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 {crab 3428 ∩ cin 3944 ⊆ wss 3945 𝒫 cpw 4598 ∪ cuni 4903 ↦ cmpt 5225 ^◡ccnv 5671 dom cdm 5672 ran crn 5673 “ cima 5675 Fun wfun 6536 Fn wfn 6537 ‘cfv 6542 (class class class)co 7414 Topctop 22788 TopOnctopon 22805 Clsdccld 22913 clsccl 22915 Cn ccn 23121 Nrmcnrm 23207 KQckq 23590

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8840 df-qtop 17482 df-top 22789 df-topon 22806 df-cld 22916 df-cls 22918 df-cn 23124 df-nrm 23214 df-kq 23591

This theorem is referenced by: kqnrm 23649

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