Theorem dif1enlem 9181

Description: Lemma for rexdif1en 9183 and dif1en 9185. (Contributed by BTernaryTau, 18-Aug-2024.) Generalize to all ordinals and add a sethood requirement to avoid ax-un 7740. (Revised by BTernaryTau, 5-Jan-2025.)

Assertion

Ref	Expression
dif1enlem	⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑀 ∈ On) ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)

Proof of Theorem dif1enlem

Step	Hyp	Ref	Expression
1		sucidg 6450	. . . . . 6 ⊢ (𝑀 ∈ On → 𝑀 ∈ suc 𝑀)
2		dff1o3 6845	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 ↔ (𝐹:𝐴–onto→suc 𝑀 ∧ Fun ◡𝐹))
3	2	simprbi 496	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → Fun ◡𝐹)
4	3	adantr 480	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → Fun ◡𝐹)
5		f1ofo 6846	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → 𝐹:𝐴–onto→suc 𝑀)
6		f1ofn 6840	. . . . . . . . . 10 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → 𝐹 Fn 𝐴)
7		fnresdm 6674	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
8		foeq1 6807	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝐴) = 𝐹 → ((𝐹 ↾ 𝐴):𝐴–onto→suc 𝑀 ↔ 𝐹:𝐴–onto→suc 𝑀))
9	6, 7, 8	3syl 18	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → ((𝐹 ↾ 𝐴):𝐴–onto→suc 𝑀 ↔ 𝐹:𝐴–onto→suc 𝑀))
10	5, 9	mpbird 257	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → (𝐹 ↾ 𝐴):𝐴–onto→suc 𝑀)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ 𝐴):𝐴–onto→suc 𝑀)
12	6	adantr 480	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → 𝐹 Fn 𝐴)
13		f1ocnvdm 7294	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝐹‘𝑀) ∈ 𝐴)
14		f1ocnvfv2 7286	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹‘(◡𝐹‘𝑀)) = 𝑀)
15		snidg 4663	. . . . . . . . . . 11 ⊢ (𝑀 ∈ suc 𝑀 → 𝑀 ∈ {𝑀})
16	15	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → 𝑀 ∈ {𝑀})
17	14, 16	eqeltrd 2829	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀})
18		fressnfv 7169	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐹‘𝑀) ∈ 𝐴) → ((𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀} ↔ (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀}))
19	18	biimp3ar 1467	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐹‘𝑀) ∈ 𝐴 ∧ (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀}) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀})
20	12, 13, 17, 19	syl3anc 1369	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀})
21		disjsn 4716	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅ ↔ ¬ (◡𝐹‘𝑀) ∈ 𝐴)
22	21	con2bii 357	. . . . . . . . . . 11 ⊢ ((◡𝐹‘𝑀) ∈ 𝐴 ↔ ¬ (𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅)
23	13, 22	sylib 217	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ¬ (𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅)
24		fnresdisj 6675	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → ((𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅ ↔ (𝐹 ↾ {(◡𝐹‘𝑀)}) = ∅))
25	6, 24	syl 17	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → ((𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅ ↔ (𝐹 ↾ {(◡𝐹‘𝑀)}) = ∅))
26	25	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ((𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅ ↔ (𝐹 ↾ {(◡𝐹‘𝑀)}) = ∅))
27	23, 26	mtbid 324	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ¬ (𝐹 ↾ {(◡𝐹‘𝑀)}) = ∅)
28	27	neqned 2944	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}) ≠ ∅)
29		foconst 6826	. . . . . . . 8 ⊢ (((𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀} ∧ (𝐹 ↾ {(◡𝐹‘𝑀)}) ≠ ∅) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}–onto→{𝑀})
30	20, 28, 29	syl2anc 583	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}–onto→{𝑀})
31		resdif 6860	. . . . . . 7 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→suc 𝑀 ∧ (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}–onto→{𝑀}) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→(suc 𝑀 ∖ {𝑀}))
32	4, 11, 30, 31	syl3anc 1369	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→(suc 𝑀 ∖ {𝑀}))
33	1, 32	sylan2 592	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ On) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→(suc 𝑀 ∖ {𝑀}))
34		eloni 6379	. . . . . . . 8 ⊢ (𝑀 ∈ On → Ord 𝑀)
35		orddif 6465	. . . . . . . 8 ⊢ (Ord 𝑀 → 𝑀 = (suc 𝑀 ∖ {𝑀}))
36	34, 35	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ On → 𝑀 = (suc 𝑀 ∖ {𝑀}))
37	36	f1oeq3d 6836	. . . . . 6 ⊢ (𝑀 ∈ On → ((𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀 ↔ (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→(suc 𝑀 ∖ {𝑀})))
38	37	adantl 481	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ On) → ((𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀 ↔ (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→(suc 𝑀 ∖ {𝑀})))
39	33, 38	mpbird 257	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ On) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀)
40	39	ancoms 458	. . 3 ⊢ ((𝑀 ∈ On ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀)
41	40	3ad2antl3 1185	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑀 ∈ On) ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀)
42		difexg 5329	. . 3 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ∖ {(◡𝐹‘𝑀)}) ∈ V)
43		resexg 6031	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})) ∈ V)
44		f1oen4g 8985	. . . 4 ⊢ ((((𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})) ∈ V ∧ (𝐴 ∖ {(◡𝐹‘𝑀)}) ∈ V ∧ 𝑀 ∈ On) ∧ (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)
45	43, 44	syl3anl1 1410	. . 3 ⊢ (((𝐹 ∈ 𝑉 ∧ (𝐴 ∖ {(◡𝐹‘𝑀)}) ∈ V ∧ 𝑀 ∈ On) ∧ (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)
46	42, 45	syl3anl2 1411	. 2 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑀 ∈ On) ∧ (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)
47	41, 46	syldan 590	1 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑀 ∈ On) ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2937  Vcvv 3471   ∖ cdif 3944   ∩ cin 3946  ∅c0 4323  {csn 4629   class class class wbr 5148  ^◡ccnv 5677   ↾ cres 5680  Ord word 6368  Oncon0 6369  suc csuc 6371  Fun wfun 6542   Fn wfn 6543  ⟶wf 6544  –onto→wfo 6546  –1-1-onto→wf1o 6547  ‘cfv 6548   ≈ cen 8961

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-sep 5299  ax-nul 5306  ax-pr 5429

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-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  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-br 5149  df-opab 5211  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  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-ord 6372  df-on 6373  df-suc 6375  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-en 8965

This theorem is referenced by:  rexdif1en  9183  dif1en  9185

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