Theorem cnneiima 47935

Description: Given a continuous function, the preimage of a neighborhood is a neighborhood. To be precise, the preimage of a neighborhood of a subset 𝑇 of the codomain of a continuous function is a neighborhood of any subset of the preimage of 𝑇. (Contributed by Zhi Wang, 9-Sep-2024.)

Hypotheses

Ref	Expression
cnneiima.1	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
cnneiima.2	⊢ (𝜑 → 𝑁 ∈ ((nei‘𝐾)‘𝑇))
cnneiima.3	⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝑇))

Assertion

Ref	Expression
cnneiima	⊢ (𝜑 → (◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆))

Proof of Theorem cnneiima

Step	Hyp	Ref	Expression
1		cnneiima.3	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝑇))
2		cnneiima.1	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
3		eqid 2728	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2728	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 23149	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	2, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶∪ 𝐾)
7	6	ffund 6726	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
8		cnneiima.2	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ((nei‘𝐾)‘𝑇))
9		cntop2 23144	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
10	2, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
11	4	neiss2 23004	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑇 ⊆ ∪ 𝐾)
12	10, 8, 11	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐾)
13	4	neii1 23009	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑁 ⊆ ∪ 𝐾)
14	10, 8, 13	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → 𝑁 ⊆ ∪ 𝐾)
15	4	neiint 23007	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝑇 ⊆ ∪ 𝐾 ∧ 𝑁 ⊆ ∪ 𝐾) → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
16	10, 12, 14, 15	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
17	8, 16	mpbid 231	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ((int‘𝐾)‘𝑁))
18		sspreima 7077	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑇 ⊆ ((int‘𝐾)‘𝑁)) → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
19	7, 17, 18	syl2anc 583	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
20	1, 19	sstrd 3990	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
21	4	cnntri 23174	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑁 ⊆ ∪ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘𝑁)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
22	2, 14, 21	syl2anc 583	. . 3 ⊢ (𝜑 → (◡𝐹 “ ((int‘𝐾)‘𝑁)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
23	20, 22	sstrd 3990	. 2 ⊢ (𝜑 → 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
24		cntop1 23143	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
25	2, 24	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
26		sspreima 7077	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑇 ⊆ ∪ 𝐾) → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ∪ 𝐾))
27	7, 12, 26	syl2anc 583	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ∪ 𝐾))
28		fimacnv 6745	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
29	6, 28	syl 17	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
30	27, 29	sseqtrd 4020	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ ∪ 𝐽)
31	1, 30	sstrd 3990	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
32		sspreima 7077	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑁 ⊆ ∪ 𝐾) → (◡𝐹 “ 𝑁) ⊆ (◡𝐹 “ ∪ 𝐾))
33	7, 14, 32	syl2anc 583	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ⊆ (◡𝐹 “ ∪ 𝐾))
34	33, 29	sseqtrd 4020	. . 3 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ⊆ ∪ 𝐽)
35	3	neiint 23007	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ (◡𝐹 “ 𝑁) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁))))
36	25, 31, 34, 35	syl3anc 1369	. 2 ⊢ (𝜑 → ((◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁))))
37	23, 36	mpbird 257	1 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1534   ∈ wcel 2099   ⊆ wss 3947  ∪ cuni 4908  ^◡ccnv 5677   “ cima 5681  Fun wfun 6542  ⟶wf 6544  ‘cfv 6548  (class class class)co 7420  Topctop 22794  intcnt 22920  neicnei 23000   Cn ccn 23127

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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

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 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8846  df-top 22795  df-topon 22812  df-ntr 22923  df-nei 23001  df-cn 23130

This theorem is referenced by:  sepfsepc  47946

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