Theorem dif1enlem 8771

Description: Lemma for rexdif1en 8772 and dif1en 8773. (Contributed by BTernaryTau, 18-Aug-2024.)

Assertion

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

Proof of Theorem dif1enlem

Step	Hyp	Ref	Expression
1		simp1 1137	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑀 ∈ ω ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → 𝐹 ∈ 𝑉)
2		sucidg 6260	. . . . . 6 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
3		dff1o3 6636	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 ↔ (𝐹:𝐴–onto→suc 𝑀 ∧ Fun ◡𝐹))
4	3	simprbi 500	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → Fun ◡𝐹)
5	4	adantr 484	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → Fun ◡𝐹)
6		f1ofo 6637	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → 𝐹:𝐴–onto→suc 𝑀)
7		f1ofn 6631	. . . . . . . . . 10 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → 𝐹 Fn 𝐴)
8		fnresdm 6465	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
9		foeq1 6598	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝐴) = 𝐹 → ((𝐹 ↾ 𝐴):𝐴–onto→suc 𝑀 ↔ 𝐹:𝐴–onto→suc 𝑀))
10	7, 8, 9	3syl 18	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → ((𝐹 ↾ 𝐴):𝐴–onto→suc 𝑀 ↔ 𝐹:𝐴–onto→suc 𝑀))
11	6, 10	mpbird 260	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → (𝐹 ↾ 𝐴):𝐴–onto→suc 𝑀)
12	11	adantr 484	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ 𝐴):𝐴–onto→suc 𝑀)
13	7	adantr 484	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → 𝐹 Fn 𝐴)
14		f1ocnvdm 7064	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝐹‘𝑀) ∈ 𝐴)
15		f1ocnvfv2 7057	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹‘(◡𝐹‘𝑀)) = 𝑀)
16		snidg 4560	. . . . . . . . . . 11 ⊢ (𝑀 ∈ suc 𝑀 → 𝑀 ∈ {𝑀})
17	16	adantl 485	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → 𝑀 ∈ {𝑀})
18	15, 17	eqeltrd 2834	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀})
19		fressnfv 6944	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐹‘𝑀) ∈ 𝐴) → ((𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀} ↔ (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀}))
20	19	biimp3ar 1471	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐹‘𝑀) ∈ 𝐴 ∧ (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀}) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀})
21	13, 14, 18, 20	syl3anc 1372	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀})
22		disjsn 4612	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅ ↔ ¬ (◡𝐹‘𝑀) ∈ 𝐴)
23	22	con2bii 361	. . . . . . . . . . 11 ⊢ ((◡𝐹‘𝑀) ∈ 𝐴 ↔ ¬ (𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅)
24	14, 23	sylib 221	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ¬ (𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅)
25		fnresdisj 6466	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → ((𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅ ↔ (𝐹 ↾ {(◡𝐹‘𝑀)}) = ∅))
26	7, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → ((𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅ ↔ (𝐹 ↾ {(◡𝐹‘𝑀)}) = ∅))
27	26	adantr 484	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ((𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅ ↔ (𝐹 ↾ {(◡𝐹‘𝑀)}) = ∅))
28	24, 27	mtbid 327	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ¬ (𝐹 ↾ {(◡𝐹‘𝑀)}) = ∅)
29	28	neqned 2942	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}) ≠ ∅)
30		foconst 6617	. . . . . . . 8 ⊢ (((𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀} ∧ (𝐹 ↾ {(◡𝐹‘𝑀)}) ≠ ∅) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}–onto→{𝑀})
31	21, 29, 30	syl2anc 587	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}–onto→{𝑀})
32		resdif 6650	. . . . . . 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}))
33	5, 12, 31, 32	syl3anc 1372	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→(suc 𝑀 ∖ {𝑀}))
34	2, 33	sylan2 596	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ ω) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→(suc 𝑀 ∖ {𝑀}))
35		nnord 7619	. . . . . . . 8 ⊢ (𝑀 ∈ ω → Ord 𝑀)
36		orddif 6275	. . . . . . . 8 ⊢ (Ord 𝑀 → 𝑀 = (suc 𝑀 ∖ {𝑀}))
37	35, 36	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 = (suc 𝑀 ∖ {𝑀}))
38	37	f1oeq3d 6627	. . . . . 6 ⊢ (𝑀 ∈ ω → ((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	38	adantl 485	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ ω) → ((𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀 ↔ (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→(suc 𝑀 ∖ {𝑀})))
40	34, 39	mpbird 260	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ ω) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀)
41	40	ancoms 462	. . 3 ⊢ ((𝑀 ∈ ω ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀)
42	41	3adant1 1131	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑀 ∈ ω ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀)
43		resexg 5881	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})) ∈ V)
44		f1oen3g 8583	. . 3 ⊢ (((𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})) ∈ V ∧ (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)
45	43, 44	sylan 583	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)
46	1, 42, 45	syl2anc 587	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑀 ∈ ω ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114   ≠ wne 2935  Vcvv 3400   ∖ cdif 3850   ∩ cin 3852  ∅c0 4221  {csn 4526   class class class wbr 5040  ^◡ccnv 5534   ↾ cres 5537  Ord word 6181  suc csuc 6184  Fun wfun 6343   Fn wfn 6344  ⟶wf 6345  –onto→wfo 6347  –1-1-onto→wf1o 6348  ‘cfv 6349  ωcom 7611   ≈ cen 8564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-ord 6185  df-on 6186  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7612  df-en 8568

This theorem is referenced by:  rexdif1en  8772  dif1en  8773