Theorem fveqf1o 7058

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, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}))

Assertion

Ref	Expression
fveqf1o	⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐺:𝐴–1-1-onto→𝐵 ∧ (𝐺‘𝐶) = 𝐷))

Proof of Theorem fveqf1o

Step	Hyp	Ref	Expression
1		simp1 1132	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
2		f1oi 6652	. . . . . . . 8 ⊢ ( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})):(𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})–1-1-onto→(𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})
3	2	a1i 11	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})):(𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})–1-1-onto→(𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}))
4		simp2 1133	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ∈ 𝐴)
5		f1ocnv 6627	. . . . . . . . . 10 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
6		f1of 6615	. . . . . . . . . 10 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴)
7	1, 5, 6	3syl 18	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ◡𝐹:𝐵⟶𝐴)
8		simp3 1134	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐷 ∈ 𝐵)
9	7, 8	ffvelrnd 6852	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (◡𝐹‘𝐷) ∈ 𝐴)
10		f1oprswap 6658	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐴 ∧ (◡𝐹‘𝐷) ∈ 𝐴) → {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}:{𝐶, (◡𝐹‘𝐷)}–1-1-onto→{𝐶, (◡𝐹‘𝐷)})
11	4, 9, 10	syl2anc 586	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}:{𝐶, (◡𝐹‘𝐷)}–1-1-onto→{𝐶, (◡𝐹‘𝐷)})
12		incom 4178	. . . . . . . . 9 ⊢ ((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∩ {𝐶, (◡𝐹‘𝐷)}) = ({𝐶, (◡𝐹‘𝐷)} ∩ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}))
13		disjdif 4421	. . . . . . . . 9 ⊢ ({𝐶, (◡𝐹‘𝐷)} ∩ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) = ∅
14	12, 13	eqtri 2844	. . . . . . . 8 ⊢ ((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∩ {𝐶, (◡𝐹‘𝐷)}) = ∅
15	14	a1i 11	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∩ {𝐶, (◡𝐹‘𝐷)}) = ∅)
16		f1oun 6634	. . . . . . 7 ⊢ (((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})):(𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})–1-1-onto→(𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∧ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}:{𝐶, (◡𝐹‘𝐷)}–1-1-onto→{𝐶, (◡𝐹‘𝐷)}) ∧ (((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∩ {𝐶, (◡𝐹‘𝐷)}) = ∅ ∧ ((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∩ {𝐶, (◡𝐹‘𝐷)}) = ∅)) → (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)})–1-1-onto→((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)}))
17	3, 11, 15, 15, 16	syl22anc 836	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)})–1-1-onto→((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)}))
18		uncom 4129	. . . . . . . 8 ⊢ ((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)}) = ({𝐶, (◡𝐹‘𝐷)} ∪ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}))
19	4, 9	prssd 4755	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → {𝐶, (◡𝐹‘𝐷)} ⊆ 𝐴)
20		undif 4430	. . . . . . . . 9 ⊢ ({𝐶, (◡𝐹‘𝐷)} ⊆ 𝐴 ↔ ({𝐶, (◡𝐹‘𝐷)} ∪ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) = 𝐴)
21	19, 20	sylib 220	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ({𝐶, (◡𝐹‘𝐷)} ∪ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) = 𝐴)
22	18, 21	syl5eq 2868	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)}) = 𝐴)
23	22	f1oeq2d 6611	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)})–1-1-onto→((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)}) ↔ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):𝐴–1-1-onto→((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)})))
24	17, 23	mpbid 234	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):𝐴–1-1-onto→((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)}))
25	22	f1oeq3d 6612	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):𝐴–1-1-onto→((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)}) ↔ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):𝐴–1-1-onto→𝐴))
26	24, 25	mpbid 234	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):𝐴–1-1-onto→𝐴)
27		f1oco 6637	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):𝐴–1-1-onto→𝐴) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})):𝐴–1-1-onto→𝐵)
28	1, 26, 27	syl2anc 586	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})):𝐴–1-1-onto→𝐵)
29		fveqf1o.1	. . . 4 ⊢ 𝐺 = (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}))
30		f1oeq1 6604	. . . 4 ⊢ (𝐺 = (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})) → (𝐺:𝐴–1-1-onto→𝐵 ↔ (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})):𝐴–1-1-onto→𝐵))
31	29, 30	ax-mp 5	. . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐵 ↔ (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})):𝐴–1-1-onto→𝐵)
32	28, 31	sylibr 236	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐺:𝐴–1-1-onto→𝐵)
33	29	fveq1i 6671	. . . 4 ⊢ (𝐺‘𝐶) = ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}))‘𝐶)
34		f1of 6615	. . . . . 6 ⊢ ((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):𝐴–1-1-onto→𝐴 → (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):𝐴⟶𝐴)
35	26, 34	syl 17	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):𝐴⟶𝐴)
36		fvco3 6760	. . . . 5 ⊢ (((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}):𝐴⟶𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}))‘𝐶) = (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})‘𝐶)))
37	35, 4, 36	syl2anc 586	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}))‘𝐶) = (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})‘𝐶)))
38	33, 37	syl5eq 2868	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐺‘𝐶) = (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})‘𝐶)))
39		fnresi 6476	. . . . . . . 8 ⊢ ( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) Fn (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})
40	39	a1i 11	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) Fn (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}))
41		f1ofn 6616	. . . . . . . 8 ⊢ ({⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}:{𝐶, (◡𝐹‘𝐷)}–1-1-onto→{𝐶, (◡𝐹‘𝐷)} → {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩} Fn {𝐶, (◡𝐹‘𝐷)})
42	11, 41	syl 17	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩} Fn {𝐶, (◡𝐹‘𝐷)})
43		prid1g 4696	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐶, (◡𝐹‘𝐷)})
44	4, 43	syl 17	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ∈ {𝐶, (◡𝐹‘𝐷)})
45		fvun2 6755	. . . . . . 7 ⊢ ((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) Fn (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∧ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩} Fn {𝐶, (◡𝐹‘𝐷)} ∧ (((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∩ {𝐶, (◡𝐹‘𝐷)}) = ∅ ∧ 𝐶 ∈ {𝐶, (◡𝐹‘𝐷)})) → ((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})‘𝐶) = ({⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}‘𝐶))
46	40, 42, 15, 44, 45	syl112anc 1370	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})‘𝐶) = ({⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}‘𝐶))
47		f1ofun 6617	. . . . . . . 8 ⊢ ({⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}:{𝐶, (◡𝐹‘𝐷)}–1-1-onto→{𝐶, (◡𝐹‘𝐷)} → Fun {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})
48	11, 47	syl 17	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → Fun {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})
49		opex 5356	. . . . . . . 8 ⊢ ⟨𝐶, (◡𝐹‘𝐷)⟩ ∈ V
50	49	prid1 4698	. . . . . . 7 ⊢ ⟨𝐶, (◡𝐹‘𝐷)⟩ ∈ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}
51		funopfv 6717	. . . . . . 7 ⊢ (Fun {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩} → (⟨𝐶, (◡𝐹‘𝐷)⟩ ∈ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩} → ({⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}‘𝐶) = (◡𝐹‘𝐷)))
52	48, 50, 51	mpisyl 21	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ({⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}‘𝐶) = (◡𝐹‘𝐷))
53	46, 52	eqtrd 2856	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})‘𝐶) = (◡𝐹‘𝐷))
54	53	fveq2d 6674	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})‘𝐶)) = (𝐹‘(◡𝐹‘𝐷)))
55		f1ocnvfv2 7034	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝐷)) = 𝐷)
56	1, 8, 55	syl2anc 586	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝐷)) = 𝐷)
57	54, 56	eqtrd 2856	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐹‘((( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})‘𝐶)) = 𝐷)
58	38, 57	eqtrd 2856	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐺‘𝐶) = 𝐷)
59	32, 58	jca 514	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐺:𝐴–1-1-onto→𝐵 ∧ (𝐺‘𝐶) = 𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  {cpr 4569  ⟨cop 4573   I cid 5459  ^◡ccnv 5554   ↾ cres 5557   ∘ ccom 5559  Fun wfun 6349   Fn wfn 6350  ⟶wf 6351  –1-1-onto→wf1o 6354  ‘cfv 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363

This theorem is referenced by:  infxpenc2  9448