Theorem fveqf1o 7248

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 1137	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
2		f1oi 6811	. . . . . . . 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 1138	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ∈ 𝐴)
5		f1ocnv 6785	. . . . . . . . . 10 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
6		f1of 6773	. . . . . . . . . 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 1139	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐷 ∈ 𝐵)
9	7, 8	ffvelcdmd 7030	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (◡𝐹‘𝐷) ∈ 𝐴)
10		f1oprswap 6818	. . . . . . . 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 585	. . . . . . 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		disjdifr 4424	. . . . . . . 8 ⊢ ((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∩ {𝐶, (◡𝐹‘𝐷)}) = ∅
13	12	a1i 11	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∩ {𝐶, (◡𝐹‘𝐷)}) = ∅)
14		f1oun 6792	. . . . . . 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)}))
15	3, 11, 13, 13, 14	syl22anc 839	. . . . . 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)}))
16		uncom 4109	. . . . . . . 8 ⊢ ((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)}) = ({𝐶, (◡𝐹‘𝐷)} ∪ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}))
17	4, 9	prssd 4777	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → {𝐶, (◡𝐹‘𝐷)} ⊆ 𝐴)
18		undif 4433	. . . . . . . . 9 ⊢ ({𝐶, (◡𝐹‘𝐷)} ⊆ 𝐴 ↔ ({𝐶, (◡𝐹‘𝐷)} ∪ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) = 𝐴)
19	17, 18	sylib 218	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ({𝐶, (◡𝐹‘𝐷)} ∪ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) = 𝐴)
20	16, 19	eqtrid 2782	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}) ∪ {𝐶, (◡𝐹‘𝐷)}) = 𝐴)
21	20	f1oeq2d 6769	. . . . . 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)})))
22	15, 21	mpbid 232	. . . . 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)}))
23	20	f1oeq3d 6770	. . . . 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))
24	22, 23	mpbid 232	. . . 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)
25		f1oco 6796	. . . 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)
26	1, 24, 25	syl2anc 585	. . 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)
27		fveqf1o.1	. . . 4 ⊢ 𝐺 = (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}))
28		f1oeq1 6761	. . . 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))
29	27, 28	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)
30	26, 29	sylibr 234	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐺:𝐴–1-1-onto→𝐵)
31	27	fveq1i 6834	. . . 4 ⊢ (𝐺‘𝐶) = ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}))‘𝐶)
32		f1of 6773	. . . . . 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)
33	24, 32	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)
34		fvco3 6932	. . . . 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)))
35	33, 4, 34	syl2anc 585	. . . 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)))
36	31, 35	eqtrid 2782	. . 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)))
37		fnresi 6620	. . . . . . . 8 ⊢ ( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) Fn (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})
38	37	a1i 11	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) Fn (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)}))
39		f1ofn 6774	. . . . . . . 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)})
40	11, 39	syl 17	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩} Fn {𝐶, (◡𝐹‘𝐷)})
41		prid1g 4716	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐶, (◡𝐹‘𝐷)})
42	4, 41	syl 17	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐶 ∈ {𝐶, (◡𝐹‘𝐷)})
43		fvun2 6925	. . . . . . 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))
44	38, 40, 13, 42, 43	syl112anc 1377	. . . . . 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))
45		f1ofun 6775	. . . . . . . 8 ⊢ ({⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}:{𝐶, (◡𝐹‘𝐷)}–1-1-onto→{𝐶, (◡𝐹‘𝐷)} → Fun {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})
46	11, 45	syl 17	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → Fun {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})
47		opex 5411	. . . . . . . 8 ⊢ ⟨𝐶, (◡𝐹‘𝐷)⟩ ∈ V
48	47	prid1 4718	. . . . . . 7 ⊢ ⟨𝐶, (◡𝐹‘𝐷)⟩ ∈ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}
49		funopfv 6882	. . . . . . 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)))
50	46, 48, 49	mpisyl 21	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ({⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩}‘𝐶) = (◡𝐹‘𝐷))
51	44, 50	eqtrd 2770	. . . . 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))
52	51	fveq2d 6837	. . . 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)))
53		f1ocnvfv2 7223	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝐷)) = 𝐷)
54	1, 8, 53	syl2anc 585	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝐷)) = 𝐷)
55	52, 54	eqtrd 2770	. . 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)
56	36, 55	eqtrd 2770	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐺‘𝐶) = 𝐷)
57	30, 56	jca 511	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∖ cdif 3897   ∪ cun 3898   ∩ cin 3899   ⊆ wss 3900  ∅c0 4284  {cpr 4581  ⟨cop 4585   I cid 5517  ^◡ccnv 5622   ↾ cres 5625   ∘ ccom 5627  Fun wfun 6485   Fn wfn 6486  ⟶wf 6487  –1-1-onto→wf1o 6490  ‘cfv 6491

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499

This theorem is referenced by:  infxpenc2  9934