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Theorem fveqf1o 7236
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 1136 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐹:𝐴1-1-onto𝐵)
2 f1oi 6810 . . . . . . . 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 1137 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐶𝐴)
5 f1ocnv 6784 . . . . . . . . . 10 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
6 f1of 6772 . . . . . . . . . 10 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
71, 5, 63syl 18 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐹:𝐵𝐴)
8 simp3 1138 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐷𝐵)
97, 8ffvelcdmd 7023 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹𝐷) ∈ 𝐴)
10 f1oprswap 6816 . . . . . . . 8 ((𝐶𝐴 ∧ (𝐹𝐷) ∈ 𝐴) → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)})
114, 9, 10syl2anc 585 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)})
12 disjdifr 4424 . . . . . . . 8 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅
1312a1i 11 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅)
14 f1oun 6791 . . . . . . 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)}))
153, 11, 13, 13, 14syl22anc 837 . . . . . 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)}))
16 uncom 4105 . . . . . . . 8 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) = ({𝐶, (𝐹𝐷)} ∪ (𝐴 ∖ {𝐶, (𝐹𝐷)}))
174, 9prssd 4774 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → {𝐶, (𝐹𝐷)} ⊆ 𝐴)
18 undif 4433 . . . . . . . . 9 ({𝐶, (𝐹𝐷)} ⊆ 𝐴 ↔ ({𝐶, (𝐹𝐷)} ∪ (𝐴 ∖ {𝐶, (𝐹𝐷)})) = 𝐴)
1917, 18sylib 217 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ({𝐶, (𝐹𝐷)} ∪ (𝐴 ∖ {𝐶, (𝐹𝐷)})) = 𝐴)
2016, 19eqtrid 2789 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) = 𝐴)
2120f1oeq2d 6768 . . . . . 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)})))
2215, 21mpbid 231 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto→((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}))
2320f1oeq3d 6769 . . . . 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))
2422, 23mpbid 231 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto𝐴)
25 f1oco 6795 . . . 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)
261, 24, 25syl2anc 585 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})):𝐴1-1-onto𝐵)
27 fveqf1o.1 . . . 4 𝐺 = (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}))
28 f1oeq1 6760 . . . 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))
2927, 28ax-mp 5 . . 3 (𝐺:𝐴1-1-onto𝐵 ↔ (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})):𝐴1-1-onto𝐵)
3026, 29sylibr 233 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐺:𝐴1-1-onto𝐵)
3127fveq1i 6831 . . . 4 (𝐺𝐶) = ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}))‘𝐶)
32 f1of 6772 . . . . . 6 ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴1-1-onto𝐴 → (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴𝐴)
3324, 32syl 17 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}):𝐴𝐴)
34 fvco3 6928 . . . . 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)))
3533, 4, 34syl2anc 585 . . . 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)))
3631, 35eqtrid 2789 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺𝐶) = (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)))
37 fnresi 6618 . . . . . . . 8 ( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) Fn (𝐴 ∖ {𝐶, (𝐹𝐷)})
3837a1i 11 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) Fn (𝐴 ∖ {𝐶, (𝐹𝐷)}))
39 f1ofn 6773 . . . . . . . 8 ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)} → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩} Fn {𝐶, (𝐹𝐷)})
4011, 39syl 17 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩} Fn {𝐶, (𝐹𝐷)})
41 prid1g 4713 . . . . . . . 8 (𝐶𝐴𝐶 ∈ {𝐶, (𝐹𝐷)})
424, 41syl 17 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐶 ∈ {𝐶, (𝐹𝐷)})
43 fvun2 6921 . . . . . . 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))
4438, 40, 13, 42, 43syl112anc 1374 . . . . . 6 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶) = ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}‘𝐶))
45 f1ofun 6774 . . . . . . . 8 ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)} → Fun {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})
4611, 45syl 17 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → Fun {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})
47 opex 5414 . . . . . . . 8 𝐶, (𝐹𝐷)⟩ ∈ V
4847prid1 4715 . . . . . . 7 𝐶, (𝐹𝐷)⟩ ∈ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}
49 funopfv 6882 . . . . . . 7 (Fun {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩} → (⟨𝐶, (𝐹𝐷)⟩ ∈ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩} → ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}‘𝐶) = (𝐹𝐷)))
5046, 48, 49mpisyl 21 . . . . . 6 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ({⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}‘𝐶) = (𝐹𝐷))
5144, 50eqtrd 2777 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶) = (𝐹𝐷))
5251fveq2d 6834 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)) = (𝐹‘(𝐹𝐷)))
53 f1ocnvfv2 7210 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐷𝐵) → (𝐹‘(𝐹𝐷)) = 𝐷)
541, 8, 53syl2anc 585 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘(𝐹𝐷)) = 𝐷)
5552, 54eqtrd 2777 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)) = 𝐷)
5636, 55eqtrd 2777 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺𝐶) = 𝐷)
5730, 56jca 513 1 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺:𝐴1-1-onto𝐵 ∧ (𝐺𝐶) = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1087   = wceq 1541  wcel 2106  cdif 3899  cun 3900  cin 3901  wss 3902  c0 4274  {cpr 4580  cop 4584   I cid 5522  ccnv 5624  cres 5627  ccom 5629  Fun wfun 6478   Fn wfn 6479  wf 6480  1-1-ontowf1o 6483  cfv 6484
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5248  ax-nul 5255  ax-pr 5377
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-br 5098  df-opab 5160  df-id 5523  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492
This theorem is referenced by:  infxpenc2  9884
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