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Theorem fveqf1o 6597
 Description: Given a bijection 𝐹, produce another bijection 𝐺 which additionally maps two specified points. (Contributed by Mario Carneiro, 30-May-2015.)
Hypothesis
Ref Expression
fveqf1o.1 𝐺 = (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}))
Assertion
Ref Expression
fveqf1o ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺:𝐴1-1-onto𝐵 ∧ (𝐺𝐶) = 𝐷))

Proof of Theorem fveqf1o
StepHypRef Expression
1 simp1 1081 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐹:𝐴1-1-onto𝐵)
2 f1oi 6212 . . . . . . . 8 ( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})):(𝐴 ∖ {𝐶, (𝐹𝐷)})–1-1-onto→(𝐴 ∖ {𝐶, (𝐹𝐷)})
32a1i 11 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})):(𝐴 ∖ {𝐶, (𝐹𝐷)})–1-1-onto→(𝐴 ∖ {𝐶, (𝐹𝐷)}))
4 simp2 1082 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐶𝐴)
5 f1ocnv 6187 . . . . . . . . . 10 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
6 f1of 6175 . . . . . . . . . 10 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
71, 5, 63syl 18 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐹:𝐵𝐴)
8 simp3 1083 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐷𝐵)
97, 8ffvelrnd 6400 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹𝐷) ∈ 𝐴)
10 f1oprswap 6218 . . . . . . . 8 ((𝐶𝐴 ∧ (𝐹𝐷) ∈ 𝐴) → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)})
114, 9, 10syl2anc 694 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)})
12 incom 3838 . . . . . . . . 9 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ({𝐶, (𝐹𝐷)} ∩ (𝐴 ∖ {𝐶, (𝐹𝐷)}))
13 disjdif 4073 . . . . . . . . 9 ({𝐶, (𝐹𝐷)} ∩ (𝐴 ∖ {𝐶, (𝐹𝐷)})) = ∅
1412, 13eqtri 2673 . . . . . . . 8 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅
1514a1i 11 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅)
16 f1oun 6194 . . . . . . 7 (((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})):(𝐴 ∖ {𝐶, (𝐹𝐷)})–1-1-onto→(𝐴 ∖ {𝐶, (𝐹𝐷)}) ∧ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)}) ∧ (((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅ ∧ ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅)) → (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)})–1-1-onto→((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}))
173, 11, 15, 15, 16syl22anc 1367 . . . . . 6 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)})–1-1-onto→((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}))
18 uncom 3790 . . . . . . . 8 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) = ({𝐶, (𝐹𝐷)} ∪ (𝐴 ∖ {𝐶, (𝐹𝐷)}))
19 prssi 4385 . . . . . . . . . 10 ((𝐶𝐴 ∧ (𝐹𝐷) ∈ 𝐴) → {𝐶, (𝐹𝐷)} ⊆ 𝐴)
204, 9, 19syl2anc 694 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → {𝐶, (𝐹𝐷)} ⊆ 𝐴)
21 undif 4082 . . . . . . . . 9 ({𝐶, (𝐹𝐷)} ⊆ 𝐴 ↔ ({𝐶, (𝐹𝐷)} ∪ (𝐴 ∖ {𝐶, (𝐹𝐷)})) = 𝐴)
2220, 21sylib 208 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ({𝐶, (𝐹𝐷)} ∪ (𝐴 ∖ {𝐶, (𝐹𝐷)})) = 𝐴)
2318, 22syl5eq 2697 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) = 𝐴)
24 f1oeq2 6166 . . . . . . 7 (((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) = 𝐴 → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)})–1-1-onto→((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) ↔ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto→((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)})))
2523, 24syl 17 . . . . . 6 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)})–1-1-onto→((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) ↔ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto→((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)})))
2617, 25mpbid 222 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto→((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}))
27 f1oeq3 6167 . . . . . 6 (((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) = 𝐴 → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto→((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) ↔ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto𝐴))
2823, 27syl 17 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto→((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) ↔ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto𝐴))
2926, 28mpbid 222 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto𝐴)
30 f1oco 6197 . . . 4 ((𝐹:𝐴1-1-onto𝐵 ∧ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto𝐴) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})):𝐴1-1-onto𝐵)
311, 29, 30syl2anc 694 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})):𝐴1-1-onto𝐵)
32 fveqf1o.1 . . . 4 𝐺 = (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}))
33 f1oeq1 6165 . . . 4 (𝐺 = (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})) → (𝐺:𝐴1-1-onto𝐵 ↔ (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})):𝐴1-1-onto𝐵))
3432, 33ax-mp 5 . . 3 (𝐺:𝐴1-1-onto𝐵 ↔ (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})):𝐴1-1-onto𝐵)
3531, 34sylibr 224 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐺:𝐴1-1-onto𝐵)
3632fveq1i 6230 . . . 4 (𝐺𝐶) = ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}))‘𝐶)
37 f1of 6175 . . . . . 6 ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto𝐴 → (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴𝐴)
3829, 37syl 17 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴𝐴)
39 fvco3 6314 . . . . 5 (((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴𝐴𝐶𝐴) → ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}))‘𝐶) = (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)))
4038, 4, 39syl2anc 694 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}))‘𝐶) = (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)))
4136, 40syl5eq 2697 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺𝐶) = (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)))
42 fnresi 6046 . . . . . . . 8 ( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) Fn (𝐴 ∖ {𝐶, (𝐹𝐷)})
4342a1i 11 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) Fn (𝐴 ∖ {𝐶, (𝐹𝐷)}))
44 f1ofn 6176 . . . . . . . 8 ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)} → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩} Fn {𝐶, (𝐹𝐷)})
4511, 44syl 17 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩} Fn {𝐶, (𝐹𝐷)})
46 prid1g 4327 . . . . . . . 8 (𝐶𝐴𝐶 ∈ {𝐶, (𝐹𝐷)})
474, 46syl 17 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐶 ∈ {𝐶, (𝐹𝐷)})
48 fvun2 6309 . . . . . . 7 ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) Fn (𝐴 ∖ {𝐶, (𝐹𝐷)}) ∧ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩} Fn {𝐶, (𝐹𝐷)} ∧ (((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅ ∧ 𝐶 ∈ {𝐶, (𝐹𝐷)})) → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶) = ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}‘𝐶))
4943, 45, 15, 47, 48syl112anc 1370 . . . . . 6 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶) = ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}‘𝐶))
50 f1ofun 6177 . . . . . . . 8 ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)} → Fun {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})
5111, 50syl 17 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → Fun {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})
52 opex 4962 . . . . . . . 8 𝐶, (𝐹𝐷)⟩ ∈ V
5352prid1 4329 . . . . . . 7 𝐶, (𝐹𝐷)⟩ ∈ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}
54 funopfv 6273 . . . . . . 7 (Fun {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩} → (⟨𝐶, (𝐹𝐷)⟩ ∈ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩} → ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}‘𝐶) = (𝐹𝐷)))
5551, 53, 54mpisyl 21 . . . . . 6 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}‘𝐶) = (𝐹𝐷))
5649, 55eqtrd 2685 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶) = (𝐹𝐷))
5756fveq2d 6233 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)) = (𝐹‘(𝐹𝐷)))
58 f1ocnvfv2 6573 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐷𝐵) → (𝐹‘(𝐹𝐷)) = 𝐷)
591, 8, 58syl2anc 694 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘(𝐹𝐷)) = 𝐷)
6057, 59eqtrd 2685 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)) = 𝐷)
6141, 60eqtrd 2685 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺𝐶) = 𝐷)
6235, 61jca 553 1 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺:𝐴1-1-onto𝐵 ∧ (𝐺𝐶) = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  {cpr 4212  ⟨cop 4216   I cid 5052  ◡ccnv 5142   ↾ cres 5145   ∘ ccom 5147  Fun wfun 5920   Fn wfn 5921  ⟶wf 5922  –1-1-onto→wf1o 5925  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934 This theorem is referenced by:  infxpenc2  8883
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