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Theorem fveqf1o 6435
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 1053 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐹:𝐴1-1-onto𝐵)
2 f1oi 6071 . . . . . . . 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 1054 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐶𝐴)
5 f1ocnv 6047 . . . . . . . . . 10 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
6 f1of 6035 . . . . . . . . . 10 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
71, 5, 63syl 18 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐹:𝐵𝐴)
8 simp3 1055 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐷𝐵)
97, 8ffvelrnd 6253 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹𝐷) ∈ 𝐴)
10 f1oprswap 6077 . . . . . . . 8 ((𝐶𝐴 ∧ (𝐹𝐷) ∈ 𝐴) → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)})
114, 9, 10syl2anc 690 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)})
12 incom 3766 . . . . . . . . 9 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ({𝐶, (𝐹𝐷)} ∩ (𝐴 ∖ {𝐶, (𝐹𝐷)}))
13 disjdif 3991 . . . . . . . . 9 ({𝐶, (𝐹𝐷)} ∩ (𝐴 ∖ {𝐶, (𝐹𝐷)})) = ∅
1412, 13eqtri 2631 . . . . . . . 8 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅
1514a1i 11 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅)
16 f1oun 6054 . . . . . . 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 1318 . . . . . 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 3718 . . . . . . . 8 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) = ({𝐶, (𝐹𝐷)} ∪ (𝐴 ∖ {𝐶, (𝐹𝐷)}))
19 prssi 4292 . . . . . . . . . 10 ((𝐶𝐴 ∧ (𝐹𝐷) ∈ 𝐴) → {𝐶, (𝐹𝐷)} ⊆ 𝐴)
204, 9, 19syl2anc 690 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → {𝐶, (𝐹𝐷)} ⊆ 𝐴)
21 undif 4000 . . . . . . . . 9 ({𝐶, (𝐹𝐷)} ⊆ 𝐴 ↔ ({𝐶, (𝐹𝐷)} ∪ (𝐴 ∖ {𝐶, (𝐹𝐷)})) = 𝐴)
2220, 21sylib 206 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ({𝐶, (𝐹𝐷)} ∪ (𝐴 ∖ {𝐶, (𝐹𝐷)})) = 𝐴)
2318, 22syl5eq 2655 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) = 𝐴)
24 f1oeq2 6026 . . . . . . 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 220 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto→((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}))
27 f1oeq3 6027 . . . . . 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 220 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto𝐴)
30 f1oco 6057 . . . 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 690 . . 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 6025 . . . 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 222 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐺:𝐴1-1-onto𝐵)
3632fveq1i 6089 . . . 4 (𝐺𝐶) = ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}))‘𝐶)
37 f1of 6035 . . . . . 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 6170 . . . . 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 690 . . . 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 2655 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺𝐶) = (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)))
42 fnresi 5908 . . . . . . . 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 6036 . . . . . . . 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 4238 . . . . . . . 8 (𝐶𝐴𝐶 ∈ {𝐶, (𝐹𝐷)})
474, 46syl 17 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐶 ∈ {𝐶, (𝐹𝐷)})
48 fvun2 6165 . . . . . . 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 1321 . . . . . 6 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶) = ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}‘𝐶))
50 f1ofun 6037 . . . . . . . 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 4853 . . . . . . . 8 𝐶, (𝐹𝐷)⟩ ∈ V
5352prid1 4240 . . . . . . 7 𝐶, (𝐹𝐷)⟩ ∈ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}
54 funopfv 6130 . . . . . . 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 2643 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶) = (𝐹𝐷))
5756fveq2d 6092 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)) = (𝐹‘(𝐹𝐷)))
58 f1ocnvfv2 6411 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐷𝐵) → (𝐹‘(𝐹𝐷)) = 𝐷)
591, 8, 58syl2anc 690 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘(𝐹𝐷)) = 𝐷)
6057, 59eqtrd 2643 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)) = 𝐷)
6141, 60eqtrd 2643 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺𝐶) = 𝐷)
6235, 61jca 552 1 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺:𝐴1-1-onto𝐵 ∧ (𝐺𝐶) = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  cdif 3536  cun 3537  cin 3538  wss 3539  c0 3873  {cpr 4126  cop 4130   I cid 4938  ccnv 5027  cres 5030  ccom 5032  Fun wfun 5784   Fn wfn 5785  wf 5786  1-1-ontowf1o 5789  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798
This theorem is referenced by:  infxpenc2  8705
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