Theorem dif1enlemOLD 9223

Description: Obsolete version of dif1enlem 9222 as of 5-Jan-2025. (Contributed by BTernaryTau, 18-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dif1enlemOLD	⊢ ((𝐹 ∈ 𝑉 ∧ 𝑀 ∈ ω ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)

Proof of Theorem dif1enlemOLD

Step	Hyp	Ref	Expression
1		simp1 1136	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑀 ∈ ω ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → 𝐹 ∈ 𝑉)
2		sucidg 6476	. . . . . 6 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
3		dff1o3 6868	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 ↔ (𝐹:𝐴–onto→suc 𝑀 ∧ Fun ◡𝐹))
4	3	simprbi 496	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → Fun ◡𝐹)
5	4	adantr 480	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → Fun ◡𝐹)
6		f1ofo 6869	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → 𝐹:𝐴–onto→suc 𝑀)
7		f1ofn 6863	. . . . . . . . . 10 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → 𝐹 Fn 𝐴)
8		fnresdm 6699	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
9		foeq1 6830	. . . . . . . . . 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 257	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → (𝐹 ↾ 𝐴):𝐴–onto→suc 𝑀)
12	11	adantr 480	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ 𝐴):𝐴–onto→suc 𝑀)
13	7	adantr 480	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → 𝐹 Fn 𝐴)
14		f1ocnvdm 7321	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝐹‘𝑀) ∈ 𝐴)
15		f1ocnvfv2 7313	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹‘(◡𝐹‘𝑀)) = 𝑀)
16		snidg 4682	. . . . . . . . . . 11 ⊢ (𝑀 ∈ suc 𝑀 → 𝑀 ∈ {𝑀})
17	16	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → 𝑀 ∈ {𝑀})
18	15, 17	eqeltrd 2844	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀})
19		fressnfv 7194	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐹‘𝑀) ∈ 𝐴) → ((𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀} ↔ (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀}))
20	19	biimp3ar 1470	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐹‘𝑀) ∈ 𝐴 ∧ (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀}) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀})
21	13, 14, 18, 20	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀})
22		disjsn 4736	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅ ↔ ¬ (◡𝐹‘𝑀) ∈ 𝐴)
23	22	con2bii 357	. . . . . . . . . . 11 ⊢ ((◡𝐹‘𝑀) ∈ 𝐴 ↔ ¬ (𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅)
24	14, 23	sylib 218	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ¬ (𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅)
25		fnresdisj 6700	. . . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ((𝐴 ∩ {(◡𝐹‘𝑀)}) = ∅ ↔ (𝐹 ↾ {(◡𝐹‘𝑀)}) = ∅))
28	24, 27	mtbid 324	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → ¬ (𝐹 ↾ {(◡𝐹‘𝑀)}) = ∅)
29	28	neqned 2953	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}) ≠ ∅)
30		foconst 6849	. . . . . . . 8 ⊢ (((𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀} ∧ (𝐹 ↾ {(◡𝐹‘𝑀)}) ≠ ∅) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}–onto→{𝑀})
31	21, 29, 30	syl2anc 583	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}–onto→{𝑀})
32		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}))
33	5, 12, 31, 32	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}))
34	2, 33	sylan2 592	. . . . 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 7911	. . . . . . . 8 ⊢ (𝑀 ∈ ω → Ord 𝑀)
36		orddif 6491	. . . . . . . 8 ⊢ (Ord 𝑀 → 𝑀 = (suc 𝑀 ∖ {𝑀}))
37	35, 36	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 = (suc 𝑀 ∖ {𝑀}))
38	37	f1oeq3d 6859	. . . . . 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 481	. . . . 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 257	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ ω) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀)
41	40	ancoms 458	. . 3 ⊢ ((𝑀 ∈ ω ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀)
42	41	3adant1 1130	. 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 6056	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})) ∈ V)
44		f1oen3g 9026	. . 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 579	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})):(𝐴 ∖ {(◡𝐹‘𝑀)})–1-1-onto→𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)
46	1, 42, 45	syl2anc 583	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑀 ∈ ω ∧ 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  suc csuc 6397  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  –onto→wfo 6571  –1-1-onto→wf1o 6572  ‘cfv 6573  ωcom 7903   ≈ 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  ax-un 7770

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-om 7904  df-en 9004

This theorem is referenced by:  rexdif1enOLD  9225  dif1enOLD  9228