Theorem dif1enlem 9222

Description: Lemma for rexdif1en 9224 and dif1en 9226. (Contributed by BTernaryTau, 18-Aug-2024.) Generalize to all ordinals and add a sethood requirement to avoid ax-un 7770. (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 6476	. . . . . 6 ⊢ (𝑀 ∈ On → 𝑀 ∈ suc 𝑀)
2		dff1o3 6868	. . . . . . . . 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 6869	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → 𝐹:𝐴–onto→suc 𝑀)
6		f1ofn 6863	. . . . . . . . . 10 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → 𝐹 Fn 𝐴)
7		fnresdm 6699	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
8		foeq1 6830	. . . . . . . . . 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 7321	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝐹‘𝑀) ∈ 𝐴)
14		f1ocnvfv2 7313	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹‘(◡𝐹‘𝑀)) = 𝑀)
15		snidg 4682	. . . . . . . . . . 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 2844	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀})
18		fressnfv 7194	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐹‘𝑀) ∈ 𝐴) → ((𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀} ↔ (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀}))
19	18	biimp3ar 1470	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐹‘𝑀) ∈ 𝐴 ∧ (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀}) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀})
20	12, 13, 17, 19	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀})
21		disjsn 4736	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅ ↔ ¬ (◡𝐹‘𝑀) ∈ 𝐴)
22	21	con2bii 357	. . . . . . . . . . 11 ⊢ ((◡𝐹‘𝑀) ∈ 𝐴 ↔ ¬ (𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅)
23	13, 22	sylib 218	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ¬ (𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅)
24		fnresdisj 6700	. . . . . . . . . . . 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 2953	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}) ≠ ∅)
29		foconst 6849	. . . . . . . 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 6883	. . . . . . 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 1371	. . . . . 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 6405	. . . . . . . 8 ⊢ (𝑀 ∈ On → Ord 𝑀)
35		orddif 6491	. . . . . . . 8 ⊢ (Ord 𝑀 → 𝑀 = (suc 𝑀 ∖ {𝑀}))
36	34, 35	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ On → 𝑀 = (suc 𝑀 ∖ {𝑀}))
37	36	f1oeq3d 6859	. . . . . 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 1187	. 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 5347	. . 3 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ∖ {(◡𝐹‘𝑀)}) ∈ V)
43		resexg 6056	. . . 4 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})) ∈ V)
44		f1oen4g 9024	. . . 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 1412	. . 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 1413	. 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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975  ∅c0 4352  {csn 4648   class class class wbr 5166  ^◡ccnv 5699   ↾ cres 5702  Ord word 6394  Oncon0 6395  suc csuc 6397  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  –onto→wfo 6571  –1-1-onto→wf1o 6572  ‘cfv 6573   ≈ cen 9000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-en 9004

This theorem is referenced by:  rexdif1en  9224  dif1en  9226