Step |
Hyp |
Ref |
Expression |
1 |
|
invisoinv.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
2 |
|
invisoinv.i |
⊢ 𝐼 = ( Iso ‘ 𝐶 ) |
3 |
|
invisoinv.n |
⊢ 𝑁 = ( Inv ‘ 𝐶 ) |
4 |
|
invisoinv.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
5 |
|
invisoinv.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
invisoinv.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
|
invisoinv.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) ) |
8 |
|
eqid |
⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) |
9 |
|
eqid |
⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 ) |
10 |
1 9 4 6
|
idiso |
⊢ ( 𝜑 → ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ∈ ( 𝑌 ( Iso ‘ 𝐶 ) 𝑌 ) ) |
11 |
2
|
a1i |
⊢ ( 𝜑 → 𝐼 = ( Iso ‘ 𝐶 ) ) |
12 |
11
|
oveqd |
⊢ ( 𝜑 → ( 𝑌 𝐼 𝑌 ) = ( 𝑌 ( Iso ‘ 𝐶 ) 𝑌 ) ) |
13 |
10 12
|
eleqtrrd |
⊢ ( 𝜑 → ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ∈ ( 𝑌 𝐼 𝑌 ) ) |
14 |
1 3 4 5 6 2 7 8 6 13
|
invco |
⊢ ( 𝜑 → ( ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑌 ) 𝐹 ) ( 𝑋 𝑁 𝑌 ) ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 〈 𝑌 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) ( ( 𝑌 𝑁 𝑌 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) ) ) |
15 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
16 |
1 15 2 4 5 6
|
isohom |
⊢ ( 𝜑 → ( 𝑋 𝐼 𝑌 ) ⊆ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) |
17 |
16 7
|
sseldd |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) |
18 |
1 15 9 4 5 8 6 17
|
catlid |
⊢ ( 𝜑 → ( ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑌 ) 𝐹 ) = 𝐹 ) |
19 |
3
|
a1i |
⊢ ( 𝜑 → 𝑁 = ( Inv ‘ 𝐶 ) ) |
20 |
19
|
oveqd |
⊢ ( 𝜑 → ( 𝑌 𝑁 𝑌 ) = ( 𝑌 ( Inv ‘ 𝐶 ) 𝑌 ) ) |
21 |
20
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝑌 𝑁 𝑌 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) = ( ( 𝑌 ( Inv ‘ 𝐶 ) 𝑌 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) ) |
22 |
1 9 4 6
|
idinv |
⊢ ( 𝜑 → ( ( 𝑌 ( Inv ‘ 𝐶 ) 𝑌 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) = ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) |
23 |
21 22
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑌 𝑁 𝑌 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) = ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) |
24 |
23
|
oveq2d |
⊢ ( 𝜑 → ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 〈 𝑌 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) ( ( 𝑌 𝑁 𝑌 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) ) = ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 〈 𝑌 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) ) |
25 |
1 15 2 4 6 5
|
isohom |
⊢ ( 𝜑 → ( 𝑌 𝐼 𝑋 ) ⊆ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
26 |
1 3 4 5 6 2
|
invf |
⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) ⟶ ( 𝑌 𝐼 𝑋 ) ) |
27 |
26 7
|
ffvelrnd |
⊢ ( 𝜑 → ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ∈ ( 𝑌 𝐼 𝑋 ) ) |
28 |
25 27
|
sseldd |
⊢ ( 𝜑 → ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
29 |
1 15 9 4 6 8 5 28
|
catrid |
⊢ ( 𝜑 → ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 〈 𝑌 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) = ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) |
30 |
24 29
|
eqtrd |
⊢ ( 𝜑 → ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 〈 𝑌 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) ( ( 𝑌 𝑁 𝑌 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑌 ) ) ) = ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) |
31 |
14 18 30
|
3brtr3d |
⊢ ( 𝜑 → 𝐹 ( 𝑋 𝑁 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) |
32 |
1 3 4 6 5
|
invsym |
⊢ ( 𝜑 → ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 𝑌 𝑁 𝑋 ) 𝐹 ↔ 𝐹 ( 𝑋 𝑁 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) ) |
33 |
31 32
|
mpbird |
⊢ ( 𝜑 → ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 𝑌 𝑁 𝑋 ) 𝐹 ) |