Metamath Proof Explorer

Theorem fveqf1o

Description: Given a bijection F , produce another bijection G which additionally maps two specified points. (Contributed by Mario Carneiro, 30-May-2015)

Ref Expression
Hypothesis fveqf1o.1 ⊢ 𝐺 = ( 𝐹 ∘ ( ( I ↾ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) ∪ { ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ , ⟨ ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 ⟩ } ) )
Assertion fveqf1o ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝐺 ‘ 𝐶 ) = 𝐷 ) )

Proof

Step Hyp Ref Expression
1 fveqf1o.1 ⊢ 𝐺 = ( 𝐹 ∘ ( ( I ↾ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) ∪ { ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ , ⟨ ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 ⟩ } ) )
2 simp1 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → 𝐹 : 𝐴 –1-1-onto→ 𝐵 )
3 f1oi ⊢ ( I ↾ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) : ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) –1-1-onto→ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } )
4 3 a1i ⊢ ( ( 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 ) } ) )
5 simp2 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → 𝐶 ∈ 𝐴 )
6 f1ocnv ⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ◡ 𝐹 : 𝐵 –1-1-onto→ 𝐴 )
7 f1of ⊢ ( ◡ 𝐹 : 𝐵 –1-1-onto→ 𝐴 → ◡ 𝐹 : 𝐵 ⟶ 𝐴 )
8 2 6 7 3syl ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ◡ 𝐹 : 𝐵 ⟶ 𝐴 )
9 simp3 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → 𝐷 ∈ 𝐵 )
10 8 9 ffvelrnd ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ 𝐷 ) ∈ 𝐴 )
11 f1oprswap ⊢ ( ( 𝐶 ∈ 𝐴 ∧ ( ◡ 𝐹 ‘ 𝐷 ) ∈ 𝐴 ) → { ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ , ⟨ ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 ⟩ } : { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } –1-1-onto→ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } )
12 5 10 11 syl2anc ⊢ ( ( 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 ) } )
13 disjdifr ⊢ ( ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ∩ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) = ∅
14 13 a1i ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ∩ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) = ∅ )
15 f1oun ⊢ ( ( ( ( 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 ) } ) )
16 4 12 14 14 15 syl22anc ⊢ ( ( 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 ) } ) )
17 uncom ⊢ ( ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ∪ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) = ( { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ∪ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) )
18 5 10 prssd ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ⊆ 𝐴 )
19 undif ⊢ ( { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ⊆ 𝐴 ↔ ( { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ∪ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) = 𝐴 )
20 18 19 sylib ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ∪ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) = 𝐴 )
21 17 20 eqtrid ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ∪ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) = 𝐴 )
22 21 f1oeq2d ⊢ ( ( 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 ) } ) ) )
23 16 22 mpbid ⊢ ( ( 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 ) } ) )
24 21 f1oeq3d ⊢ ( ( 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 ) )
25 23 24 mpbid ⊢ ( ( 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 )
26 f1oco ⊢ ( ( 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 )
27 2 25 26 syl2anc ⊢ ( ( 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 )
28 f1oeq1 ⊢ ( 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 1 28 ax-mp ⊢ ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 ∘ ( ( I ↾ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) ∪ { ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ , ⟨ ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 ⟩ } ) ) : 𝐴 –1-1-onto→ 𝐵 )
30 27 29 sylibr ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → 𝐺 : 𝐴 –1-1-onto→ 𝐵 )
31 1 fveq1i ⊢ ( 𝐺 ‘ 𝐶 ) = ( ( 𝐹 ∘ ( ( I ↾ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) ∪ { ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ , ⟨ ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 ⟩ } ) ) ‘ 𝐶 )
32 f1of ⊢ ( ( ( 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 25 32 syl ⊢ ( ( 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 ⊢ ( ( ( ( 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 5 34 syl2anc ⊢ ( ( 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 ⊢ ( ( 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 ⊢ ( I ↾ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) Fn ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } )
38 37 a1i ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( I ↾ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) Fn ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) )
39 f1ofn ⊢ ( { ⟨ 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 12 39 syl ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → { ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ , ⟨ ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 ⟩ } Fn { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } )
41 prid1g ⊢ ( 𝐶 ∈ 𝐴 → 𝐶 ∈ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } )
42 5 41 syl ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → 𝐶 ∈ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } )
43 fvun2 ⊢ ( ( ( 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 14 42 43 syl112anc ⊢ ( ( 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 ⊢ ( { ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ , ⟨ ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 ⟩ } : { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } –1-1-onto→ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } → Fun { ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ , ⟨ ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 ⟩ } )
46 12 45 syl ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → Fun { ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ , ⟨ ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 ⟩ } )
47 opex ⊢ ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ ∈ V
48 47 prid1 ⊢ ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ ∈ { ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ , ⟨ ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 ⟩ }
49 funopfv ⊢ ( 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 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( { ⟨ 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) ⟩ , ⟨ ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 ⟩ } ‘ 𝐶 ) = ( ◡ 𝐹 ‘ 𝐷 ) )
51 44 50 eqtrd ⊢ ( ( 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 ⊢ ( ( 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 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝐷 ) ) = 𝐷 )
54 2 9 53 syl2anc ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝐷 ) ) = 𝐷 )
55 52 54 eqtrd ⊢ ( ( 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 ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐺 ‘ 𝐶 ) = 𝐷 )
57 30 56 jca ⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝐺 ‘ 𝐶 ) = 𝐷 ) )