MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fveqf1o Structured version   Visualization version   GIF version

Theorem fveqf1o 7303
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 6871 . . . . . . . 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 6845 . . . . . . . . . 10 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
6 f1of 6833 . . . . . . . . . 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 7087 . . . . . . . 8 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹𝐷) ∈ 𝐴)
10 f1oprswap 6877 . . . . . . . 8 ((𝐶𝐴 ∧ (𝐹𝐷) ∈ 𝐴) → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)})
114, 9, 10syl2anc 584 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}:{𝐶, (𝐹𝐷)}–1-1-onto→{𝐶, (𝐹𝐷)})
12 disjdifr 4472 . . . . . . . 8 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅
1312a1i 11 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∩ {𝐶, (𝐹𝐷)}) = ∅)
14 f1oun 6852 . . . . . . 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 4153 . . . . . . . 8 ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) = ({𝐶, (𝐹𝐷)} ∪ (𝐴 ∖ {𝐶, (𝐹𝐷)}))
174, 9prssd 4825 . . . . . . . . 9 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → {𝐶, (𝐹𝐷)} ⊆ 𝐴)
18 undif 4481 . . . . . . . . 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 2784 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐴 ∖ {𝐶, (𝐹𝐷)}) ∪ {𝐶, (𝐹𝐷)}) = 𝐴)
2120f1oeq2d 6829 . . . . . 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 6830 . . . . 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 6856 . . . 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 584 . . 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 6821 . . . 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 6892 . . . 4 (𝐺𝐶) = ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}))‘𝐶)
32 f1of 6833 . . . . . 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 6990 . . . . 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 584 . . . 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 2784 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺𝐶) = (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)))
37 fnresi 6679 . . . . . . . 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 6834 . . . . . . . 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 4764 . . . . . . . 8 (𝐶𝐴𝐶 ∈ {𝐶, (𝐹𝐷)})
424, 41syl 17 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → 𝐶 ∈ {𝐶, (𝐹𝐷)})
43 fvun2 6983 . . . . . . 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 6835 . . . . . . . 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 5464 . . . . . . . 8 𝐶, (𝐹𝐷)⟩ ∈ V
4847prid1 4766 . . . . . . 7 𝐶, (𝐹𝐷)⟩ ∈ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}
49 funopfv 6943 . . . . . . 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 2772 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶) = (𝐹𝐷))
5251fveq2d 6895 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)) = (𝐹‘(𝐹𝐷)))
53 f1ocnvfv2 7277 . . . . 5 ((𝐹:𝐴1-1-onto𝐵𝐷𝐵) → (𝐹‘(𝐹𝐷)) = 𝐷)
541, 8, 53syl2anc 584 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘(𝐹𝐷)) = 𝐷)
5552, 54eqtrd 2772 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩})‘𝐶)) = 𝐷)
5636, 55eqtrd 2772 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺𝐶) = 𝐷)
5730, 56jca 512 1 ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺:𝐴1-1-onto𝐵 ∧ (𝐺𝐶) = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cdif 3945  cun 3946  cin 3947  wss 3948  c0 4322  {cpr 4630  cop 4634   I cid 5573  ccnv 5675  cres 5678  ccom 5680  Fun wfun 6537   Fn wfn 6538  wf 6539  1-1-ontowf1o 6542  cfv 6543
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551
This theorem is referenced by:  infxpenc2  10019
  Copyright terms: Public domain W3C validator