Theorem dif1enlem 8736

Description: Lemma for rexdif1en 8737 and dif1en 8738. (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 1133	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑀 ∈ ω ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → 𝐹 ∈ 𝑉)
2		sucidg 6251	. . . . . 6 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
3		dff1o3 6612	. . . . . . . . 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 6613	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → 𝐹:𝐴–onto→suc 𝑀)
7		f1ofn 6607	. . . . . . . . . 10 ⊢ (𝐹:𝐴–1-1-onto→suc 𝑀 → 𝐹 Fn 𝐴)
8		fnresdm 6453	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
9		foeq1 6576	. . . . . . . . . 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 7038	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝐹‘𝑀) ∈ 𝐴)
15		f1ocnvfv2 7031	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹‘(◡𝐹‘𝑀)) = 𝑀)
16		snidg 4559	. . . . . . . . . . 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 2852	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀})
19		fressnfv 6918	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐹‘𝑀) ∈ 𝐴) → ((𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀} ↔ (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀}))
20	19	biimp3ar 1467	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐹‘𝑀) ∈ 𝐴 ∧ (𝐹‘(◡𝐹‘𝑀)) ∈ {𝑀}) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀})
21	13, 14, 18, 20	syl3anc 1368	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}):{(◡𝐹‘𝑀)}⟶{𝑀})
22		disjsn 4607	. . . . . . . . . . . 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 6454	. . . . . . . . . . . 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 2958	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝐹 ↾ {(◡𝐹‘𝑀)}) ≠ ∅)
30		foconst 6593	. . . . . . . 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 6626	. . . . . . 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 1368	. . . . . 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 595	. . . . 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 7592	. . . . . . . 8 ⊢ (𝑀 ∈ ω → Ord 𝑀)
36		orddif 6266	. . . . . . . 8 ⊢ (Ord 𝑀 → 𝑀 = (suc 𝑀 ∖ {𝑀}))
37	35, 36	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 = (suc 𝑀 ∖ {𝑀}))
38	37	f1oeq3d 6603	. . . . . 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 1127	. 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 5873	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ (𝐴 ∖ {(◡𝐹‘𝑀)})) ∈ V)
44		f1oen3g 8548	. . 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 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  Vcvv 3409   ∖ cdif 3857   ∩ cin 3859  ∅c0 4227  {csn 4525   class class class wbr 5035  ^◡ccnv 5526   ↾ cres 5529  Ord word 6172  suc csuc 6175  Fun wfun 6333   Fn wfn 6334  ⟶wf 6335  –onto→wfo 6337  –1-1-onto→wf1o 6338  ‘cfv 6339  ωcom 7584   ≈ cen 8529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-ord 6176  df-on 6177  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7585  df-en 8533

This theorem is referenced by:  rexdif1en  8737  dif1en  8738