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 |
|
incom |
⊢ ( ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ∩ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) = ( { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ∩ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) |
14 |
|
disjdif |
⊢ ( { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ∩ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) = ∅ |
15 |
13 14
|
eqtri |
⊢ ( ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ∩ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) = ∅ |
16 |
15
|
a1i |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ∩ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) = ∅ ) |
17 |
|
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 ) } ) ) |
18 |
4 12 16 16 17
|
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 ) } ) ) |
19 |
|
uncom |
⊢ ( ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ∪ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) = ( { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ∪ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) |
20 |
5 10
|
prssd |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ⊆ 𝐴 ) |
21 |
|
undif |
⊢ ( { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ⊆ 𝐴 ↔ ( { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ∪ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) = 𝐴 ) |
22 |
20 21
|
sylib |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ∪ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) = 𝐴 ) |
23 |
19 22
|
syl5eq |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ∪ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) = 𝐴 ) |
24 |
23
|
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 ) } ) ) ) |
25 |
18 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 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ∪ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) |
26 |
23
|
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 ) ) |
27 |
25 26
|
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 ) |
28 |
|
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 ) |
29 |
2 27 28
|
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 ) |
30 |
|
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 ) ) |
31 |
1 30
|
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 ) |
32 |
29 31
|
sylibr |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → 𝐺 : 𝐴 –1-1-onto→ 𝐵 ) |
33 |
1
|
fveq1i |
⊢ ( 𝐺 ‘ 𝐶 ) = ( ( 𝐹 ∘ ( ( I ↾ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) ∪ { 〈 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) 〉 , 〈 ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 〉 } ) ) ‘ 𝐶 ) |
34 |
|
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 ) |
35 |
27 34
|
syl |
⊢ ( ( 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 |
⊢ ( ( ( ( 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 5 36
|
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 ) ) ) |
38 |
33 37
|
syl5eq |
⊢ ( ( 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 |
⊢ ( I ↾ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) Fn ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) |
40 |
39
|
a1i |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( I ↾ ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) Fn ( 𝐴 ∖ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) ) |
41 |
|
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 ) } ) |
42 |
12 41
|
syl |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → { 〈 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) 〉 , 〈 ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 〉 } Fn { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) |
43 |
|
prid1g |
⊢ ( 𝐶 ∈ 𝐴 → 𝐶 ∈ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) |
44 |
5 43
|
syl |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → 𝐶 ∈ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } ) |
45 |
|
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 ) ) |
46 |
40 42 16 44 45
|
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 ) ) |
47 |
|
f1ofun |
⊢ ( { 〈 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) 〉 , 〈 ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 〉 } : { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } –1-1-onto→ { 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) } → Fun { 〈 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) 〉 , 〈 ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 〉 } ) |
48 |
12 47
|
syl |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → Fun { 〈 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) 〉 , 〈 ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 〉 } ) |
49 |
|
opex |
⊢ 〈 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) 〉 ∈ V |
50 |
49
|
prid1 |
⊢ 〈 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) 〉 ∈ { 〈 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) 〉 , 〈 ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 〉 } |
51 |
|
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 ) ) ) |
52 |
48 50 51
|
mpisyl |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( { 〈 𝐶 , ( ◡ 𝐹 ‘ 𝐷 ) 〉 , 〈 ( ◡ 𝐹 ‘ 𝐷 ) , 𝐶 〉 } ‘ 𝐶 ) = ( ◡ 𝐹 ‘ 𝐷 ) ) |
53 |
46 52
|
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 ) ) |
54 |
53
|
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 ) ) ) |
55 |
|
f1ocnvfv2 |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝐷 ) ) = 𝐷 ) |
56 |
2 9 55
|
syl2anc |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝐷 ) ) = 𝐷 ) |
57 |
54 56
|
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 ) |
58 |
38 57
|
eqtrd |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐺 ‘ 𝐶 ) = 𝐷 ) |
59 |
32 58
|
jca |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐺 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝐺 ‘ 𝐶 ) = 𝐷 ) ) |