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Theorem enfixsn 9109

Description: Given two equipollent sets, a bijection can always be chosen which fixes a single point. (Contributed by Stefan O'Rear, 9-Jul-2015.)

**Assertion**
Ref	Expression
enfixsn	⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) → ∃𝑓(𝑓:𝑋–1-1-onto→𝑌 ∧ (𝑓‘𝐴) = 𝐵))

Distinct variable groups: 𝐴,𝑓 𝐵,𝑓 𝑓,𝑋 𝑓,𝑌

Proof of Theorem enfixsn Dummy variables 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1156	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) → 𝑋 ≈ 𝑌)
2		bren 8987	. . 3 ⊢ (𝑋 ≈ 𝑌 ↔ ∃𝑔 𝑔:𝑋–1-1-onto→𝑌)
3	1, 2	sylib 221	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) → ∃𝑔 𝑔:𝑋–1-1-onto→𝑌)
4		relen 8982	. . . . . . . 8 ⊢ Rel ≈
5	4	brrelex2i 5707	. . . . . . 7 ⊢ (𝑋 ≈ 𝑌 → 𝑌 ∈ V)
6	5	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) → 𝑌 ∈ V)
7	6	adantr 486	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ 𝑔:𝑋–1-1-onto→𝑌) → 𝑌 ∈ V)
8		f1of 6825	. . . . . . 7 ⊢ (𝑔:𝑋–1-1-onto→𝑌 → 𝑔:𝑋⟶𝑌)
9	8	adantl 487	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ 𝑔:𝑋–1-1-onto→𝑌) → 𝑔:𝑋⟶𝑌)
10		simpl1 1210	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ 𝑔:𝑋–1-1-onto→𝑌) → 𝐴 ∈ 𝑋)
11	9, 10	ffvelcdmd 7086	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ 𝑔:𝑋–1-1-onto→𝑌) → (𝑔‘𝐴) ∈ 𝑌)
12		simpl2 1211	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ 𝑔:𝑋–1-1-onto→𝑌) → 𝐵 ∈ 𝑌)
13		difsnen 9082	. . . . 5 ⊢ ((𝑌 ∈ V ∧ (𝑔‘𝐴) ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝑌 ∖ {(𝑔‘𝐴)}) ≈ (𝑌 ∖ {𝐵}))
14	7, 11, 12, 13	syl3anc 1398	. . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ 𝑔:𝑋–1-1-onto→𝑌) → (𝑌 ∖ {(𝑔‘𝐴)}) ≈ (𝑌 ∖ {𝐵}))
15		bren 8987	. . . 4 ⊢ ((𝑌 ∖ {(𝑔‘𝐴)}) ≈ (𝑌 ∖ {𝐵}) ↔ ∃ℎ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))
16	14, 15	sylib 221	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ 𝑔:𝑋–1-1-onto→𝑌) → ∃ℎ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))
17		fvex 6899	. . . . . . . . . . 11 ⊢ (𝑔‘𝐴) ∈ V
18	17	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → (𝑔‘𝐴) ∈ V)
19		simpl2 1211	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → 𝐵 ∈ 𝑌)
20		f1osng 6868	. . . . . . . . . 10 ⊢ (((𝑔‘𝐴) ∈ V ∧ 𝐵 ∈ 𝑌) → {⟨(𝑔‘𝐴), 𝐵⟩}:{(𝑔‘𝐴)}–1-1-onto→{𝐵})
21	18, 19, 20	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → {⟨(𝑔‘𝐴), 𝐵⟩}:{(𝑔‘𝐴)}–1-1-onto→{𝐵})
22		simprr 785	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))
23		disjdif 4425	. . . . . . . . . 10 ⊢ ({(𝑔‘𝐴)} ∩ (𝑌 ∖ {(𝑔‘𝐴)})) = ∅
24	23	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({(𝑔‘𝐴)} ∩ (𝑌 ∖ {(𝑔‘𝐴)})) = ∅)
25		disjdif 4425	. . . . . . . . . 10 ⊢ ({𝐵} ∩ (𝑌 ∖ {𝐵})) = ∅
26	25	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({𝐵} ∩ (𝑌 ∖ {𝐵})) = ∅)
27		f1oun 6845	. . . . . . . . 9 ⊢ ((({⟨(𝑔‘𝐴), 𝐵⟩}:{(𝑔‘𝐴)}–1-1-onto→{𝐵} ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵})) ∧ (({(𝑔‘𝐴)} ∩ (𝑌 ∖ {(𝑔‘𝐴)})) = ∅ ∧ ({𝐵} ∩ (𝑌 ∖ {𝐵})) = ∅)) → ({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ):({(𝑔‘𝐴)} ∪ (𝑌 ∖ {(𝑔‘𝐴)}))–1-1-onto→({𝐵} ∪ (𝑌 ∖ {𝐵})))
28	21, 22, 24, 26, 27	syl22anc 852	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ):({(𝑔‘𝐴)} ∪ (𝑌 ∖ {(𝑔‘𝐴)}))–1-1-onto→({𝐵} ∪ (𝑌 ∖ {𝐵})))
29	8	ad2antrl 741	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → 𝑔:𝑋⟶𝑌)
30		simpl1 1210	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → 𝐴 ∈ 𝑋)
31	29, 30	ffvelcdmd 7086	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → (𝑔‘𝐴) ∈ 𝑌)
32		uncom 4157	. . . . . . . . . . 11 ⊢ ({(𝑔‘𝐴)} ∪ (𝑌 ∖ {(𝑔‘𝐴)})) = ((𝑌 ∖ {(𝑔‘𝐴)}) ∪ {(𝑔‘𝐴)})
33		difsnid 4767	. . . . . . . . . . 11 ⊢ ((𝑔‘𝐴) ∈ 𝑌 → ((𝑌 ∖ {(𝑔‘𝐴)}) ∪ {(𝑔‘𝐴)}) = 𝑌)
34	32, 33	eqtrid 2807	. . . . . . . . . 10 ⊢ ((𝑔‘𝐴) ∈ 𝑌 → ({(𝑔‘𝐴)} ∪ (𝑌 ∖ {(𝑔‘𝐴)})) = 𝑌)
35	31, 34	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({(𝑔‘𝐴)} ∪ (𝑌 ∖ {(𝑔‘𝐴)})) = 𝑌)
36		uncom 4157	. . . . . . . . . . 11 ⊢ ({𝐵} ∪ (𝑌 ∖ {𝐵})) = ((𝑌 ∖ {𝐵}) ∪ {𝐵})
37		difsnid 4767	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝑌 → ((𝑌 ∖ {𝐵}) ∪ {𝐵}) = 𝑌)
38	36, 37	eqtrid 2807	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑌 → ({𝐵} ∪ (𝑌 ∖ {𝐵})) = 𝑌)
39	19, 38	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({𝐵} ∪ (𝑌 ∖ {𝐵})) = 𝑌)
40		f1oeq23 6816	. . . . . . . . 9 ⊢ ((({(𝑔‘𝐴)} ∪ (𝑌 ∖ {(𝑔‘𝐴)})) = 𝑌 ∧ ({𝐵} ∪ (𝑌 ∖ {𝐵})) = 𝑌) → (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ):({(𝑔‘𝐴)} ∪ (𝑌 ∖ {(𝑔‘𝐴)}))–1-1-onto→({𝐵} ∪ (𝑌 ∖ {𝐵})) ↔ ({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ):𝑌–1-1-onto→𝑌))
41	35, 39, 40	syl2anc 596	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ):({(𝑔‘𝐴)} ∪ (𝑌 ∖ {(𝑔‘𝐴)}))–1-1-onto→({𝐵} ∪ (𝑌 ∖ {𝐵})) ↔ ({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ):𝑌–1-1-onto→𝑌))
42	28, 41	mpbid 235	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ):𝑌–1-1-onto→𝑌)
43		simprl 783	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → 𝑔:𝑋–1-1-onto→𝑌)
44		f1oco 6849	. . . . . . 7 ⊢ ((({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ):𝑌–1-1-onto→𝑌 ∧ 𝑔:𝑋–1-1-onto→𝑌) → (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔):𝑋–1-1-onto→𝑌)
45	42, 43, 44	syl2anc 596	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔):𝑋–1-1-onto→𝑌)
46		f1ofn 6826	. . . . . . . . 9 ⊢ (𝑔:𝑋–1-1-onto→𝑌 → 𝑔 Fn 𝑋)
47	46	ad2antrl 741	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → 𝑔 Fn 𝑋)
48		fvco2 6983	. . . . . . . 8 ⊢ ((𝑔 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → ((({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔)‘𝐴) = (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ)‘(𝑔‘𝐴)))
49	47, 30, 48	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ((({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔)‘𝐴) = (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ)‘(𝑔‘𝐴)))
50		f1ofn 6826	. . . . . . . . 9 ⊢ ({⟨(𝑔‘𝐴), 𝐵⟩}:{(𝑔‘𝐴)}–1-1-onto→{𝐵} → {⟨(𝑔‘𝐴), 𝐵⟩} Fn {(𝑔‘𝐴)})
51	21, 50	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → {⟨(𝑔‘𝐴), 𝐵⟩} Fn {(𝑔‘𝐴)})
52		f1ofn 6826	. . . . . . . . 9 ⊢ (ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}) → ℎ Fn (𝑌 ∖ {(𝑔‘𝐴)}))
53	52	ad2antll 742	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ℎ Fn (𝑌 ∖ {(𝑔‘𝐴)}))
54	17	snid 4619	. . . . . . . . 9 ⊢ (𝑔‘𝐴) ∈ {(𝑔‘𝐴)}
55	54	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → (𝑔‘𝐴) ∈ {(𝑔‘𝐴)})
56		fvun1 6977	. . . . . . . 8 ⊢ (({⟨(𝑔‘𝐴), 𝐵⟩} Fn {(𝑔‘𝐴)} ∧ ℎ Fn (𝑌 ∖ {(𝑔‘𝐴)}) ∧ (({(𝑔‘𝐴)} ∩ (𝑌 ∖ {(𝑔‘𝐴)})) = ∅ ∧ (𝑔‘𝐴) ∈ {(𝑔‘𝐴)})) → (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ)‘(𝑔‘𝐴)) = ({⟨(𝑔‘𝐴), 𝐵⟩}‘(𝑔‘𝐴)))
57	51, 53, 24, 55, 56	syl112anc 1401	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ)‘(𝑔‘𝐴)) = ({⟨(𝑔‘𝐴), 𝐵⟩}‘(𝑔‘𝐴)))
58		fvsng 7186	. . . . . . . 8 ⊢ (((𝑔‘𝐴) ∈ V ∧ 𝐵 ∈ 𝑌) → ({⟨(𝑔‘𝐴), 𝐵⟩}‘(𝑔‘𝐴)) = 𝐵)
59	18, 19, 58	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({⟨(𝑔‘𝐴), 𝐵⟩}‘(𝑔‘𝐴)) = 𝐵)
60	49, 57, 59	3eqtrd 2799	. . . . . 6 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ((({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔)‘𝐴) = 𝐵)
61		snex 5396	. . . . . . . . 9 ⊢ {⟨(𝑔‘𝐴), 𝐵⟩} ∈ V
62		vex 3454	. . . . . . . . 9 ⊢ ℎ ∈ V
63	61, 62	unex 7762	. . . . . . . 8 ⊢ ({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∈ V
64		vex 3454	. . . . . . . 8 ⊢ 𝑔 ∈ V
65	63, 64	coex 7943	. . . . . . 7 ⊢ (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔) ∈ V
66		f1oeq1 6813	. . . . . . . 8 ⊢ (𝑓 = (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔) → (𝑓:𝑋–1-1-onto→𝑌 ↔ (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔):𝑋–1-1-onto→𝑌))
67		fveq1 6885	. . . . . . . . 9 ⊢ (𝑓 = (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔) → (𝑓‘𝐴) = ((({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔)‘𝐴))
68	67	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑓 = (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔) → ((𝑓‘𝐴) = 𝐵 ↔ ((({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔)‘𝐴) = 𝐵))
69	66, 68	anbi12d 644	. . . . . . 7 ⊢ (𝑓 = (({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔) → ((𝑓:𝑋–1-1-onto→𝑌 ∧ (𝑓‘𝐴) = 𝐵) ↔ ((({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔):𝑋–1-1-onto→𝑌 ∧ ((({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔)‘𝐴) = 𝐵)))
70	65, 69	spcev 3560	. . . . . 6 ⊢ (((({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔):𝑋–1-1-onto→𝑌 ∧ ((({⟨(𝑔‘𝐴), 𝐵⟩} ∪ ℎ) ∘ 𝑔)‘𝐴) = 𝐵) → ∃𝑓(𝑓:𝑋–1-1-onto→𝑌 ∧ (𝑓‘𝐴) = 𝐵))
71	45, 60, 70	syl2anc 596	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ (𝑔:𝑋–1-1-onto→𝑌 ∧ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ∃𝑓(𝑓:𝑋–1-1-onto→𝑌 ∧ (𝑓‘𝐴) = 𝐵))
72	71	expr 462	. . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ 𝑔:𝑋–1-1-onto→𝑌) → (ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}) → ∃𝑓(𝑓:𝑋–1-1-onto→𝑌 ∧ (𝑓‘𝐴) = 𝐵)))
73	72	exlimdv 1965	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ 𝑔:𝑋–1-1-onto→𝑌) → (∃ℎ ℎ:(𝑌 ∖ {(𝑔‘𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}) → ∃𝑓(𝑓:𝑋–1-1-onto→𝑌 ∧ (𝑓‘𝐴) = 𝐵)))
74	16, 73	mpd 16	. 2 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) ∧ 𝑔:𝑋–1-1-onto→𝑌) → ∃𝑓(𝑓:𝑋–1-1-onto→𝑌 ∧ (𝑓‘𝐴) = 𝐵))
75	3, 74	exlimddv 1967	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑋 ≈ 𝑌) → ∃𝑓(𝑓:𝑋–1-1-onto→𝑌 ∧ (𝑓‘𝐴) = 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∃wex 1811 ∈ wcel 2144 Vcvv 3450 ∖ cdif 4045 ∩ cin 4089 ∪ cun 4149 ∅c0 4278 {csn 4584 ⟨cop 4834 class class class wbr 5102 ∘ ccom 5654 Fn wfn 6534 ⟶wf 6535 –1-1-onto→wf1o 6538 ‘cfv 6539 ≈ cen 8974

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-en 8978

This theorem is used by: mapfien2 9405

W3C validator