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Theorem fveqf1o 6542
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 1059 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐹:𝐴1-1-onto𝐵)
2 f1oi 6161 . . . . . . . 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 1060 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐶𝐴)
5 f1ocnv 6136 . . . . . . . . . 10 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
6 f1of 6124 . . . . . . . . . 10 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
71, 5, 63syl 18 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐹:𝐵𝐴)
8 simp3 1061 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐷𝐵)
97, 8ffvelrnd 6346 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹𝐷) ∈ 𝐴)
10 f1oprswap 6167 . . . . . . . 8 ((𝐶𝐴 ∧ (𝐹𝐷) ∈ 𝐴) → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)})
114, 9, 10syl2anc 692 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)})
12 incom 3797 . . . . . . . . 9 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ({𝐶, (𝐹𝐷)} ∩ (𝐴 ∖ {𝐶, (𝐹𝐷)}))
13 disjdif 4031 . . . . . . . . 9 ({𝐶, (𝐹𝐷)} ∩ (𝐴 ∖ {𝐶, (𝐹𝐷)})) = ∅
1412, 13eqtri 2642 . . . . . . . 8 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅
1514a1i 11 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅)
16 f1oun 6143 . . . . . . 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 1325 . . . . . 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 3749 . . . . . . . 8 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) = ({𝐶, (𝐹𝐷)} ∪ (𝐴 ∖ {𝐶, (𝐹𝐷)}))
19 prssi 4344 . . . . . . . . . 10 ((𝐶𝐴 ∧ (𝐹𝐷) ∈ 𝐴) → {𝐶, (𝐹𝐷)} ⊆ 𝐴)
204, 9, 19syl2anc 692 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → {𝐶, (𝐹𝐷)} ⊆ 𝐴)
21 undif 4040 . . . . . . . . 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 2666 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) = 𝐴)
24 f1oeq2 6115 . . . . . . 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 6116 . . . . . 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 6146 . . . 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 692 . . 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 6114 . . . 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 6179 . . . 4 (𝐺𝐶) = ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}))‘𝐶)
37 f1of 6124 . . . . . 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 6262 . . . . 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 692 . . . 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 2666 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺𝐶) = (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)))
42 fnresi 5996 . . . . . . . 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 6125 . . . . . . . 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 4286 . . . . . . . 8 (𝐶𝐴𝐶 ∈ {𝐶, (𝐹𝐷)})
474, 46syl 17 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐶 ∈ {𝐶, (𝐹𝐷)})
48 fvun2 6257 . . . . . . 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 1328 . . . . . 6 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶) = ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}‘𝐶))
50 f1ofun 6126 . . . . . . . 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 4923 . . . . . . . 8 𝐶, (𝐹𝐷)⟩ ∈ V
5352prid1 4288 . . . . . . 7 𝐶, (𝐹𝐷)⟩ ∈ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}
54 funopfv 6222 . . . . . . 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 2654 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶) = (𝐹𝐷))
5756fveq2d 6182 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)) = (𝐹‘(𝐹𝐷)))
58 f1ocnvfv2 6518 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐷𝐵) → (𝐹‘(𝐹𝐷)) = 𝐷)
591, 8, 58syl2anc 692 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘(𝐹𝐷)) = 𝐷)
6057, 59eqtrd 2654 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)) = 𝐷)
6141, 60eqtrd 2654 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺𝐶) = 𝐷)
6235, 61jca 554 1 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺:𝐴1-1-onto𝐵 ∧ (𝐺𝐶) = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  cdif 3564  cun 3565  cin 3566  wss 3567  c0 3907  {cpr 4170  cop 4174   I cid 5013  ccnv 5103  cres 5106  ccom 5108  Fun wfun 5870   Fn wfn 5871  wf 5872  1-1-ontowf1o 5875  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884
This theorem is referenced by:  infxpenc2  8830
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