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 |
|
invcoisoid.1 |
⊢ 1 = ( Id ‘ 𝐶 ) |
9 |
|
invcoisoid.o |
⊢ ⚬ = ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) |
10 |
1 2 3 4 5 6 7
|
invisoinvr |
⊢ ( 𝜑 → 𝐹 ( 𝑋 𝑁 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) |
11 |
|
eqid |
⊢ ( Sect ‘ 𝐶 ) = ( Sect ‘ 𝐶 ) |
12 |
1 3 4 5 6 11
|
isinv |
⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑁 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ↔ ( 𝐹 ( 𝑋 ( Sect ‘ 𝐶 ) 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ∧ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) 𝐹 ) ) ) |
13 |
|
simpl |
⊢ ( ( 𝐹 ( 𝑋 ( Sect ‘ 𝐶 ) 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ∧ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) 𝐹 ) → 𝐹 ( 𝑋 ( Sect ‘ 𝐶 ) 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) |
14 |
12 13
|
syl6bi |
⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑁 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) → 𝐹 ( 𝑋 ( Sect ‘ 𝐶 ) 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) ) |
15 |
10 14
|
mpd |
⊢ ( 𝜑 → 𝐹 ( 𝑋 ( Sect ‘ 𝐶 ) 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) |
16 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
17 |
|
eqid |
⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) |
18 |
1 16 2 4 5 6
|
isohom |
⊢ ( 𝜑 → ( 𝑋 𝐼 𝑌 ) ⊆ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) |
19 |
18 7
|
sseldd |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) |
20 |
1 16 2 4 6 5
|
isohom |
⊢ ( 𝜑 → ( 𝑌 𝐼 𝑋 ) ⊆ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
21 |
1 3 4 5 6 2
|
invf |
⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) ⟶ ( 𝑌 𝐼 𝑋 ) ) |
22 |
21 7
|
ffvelrnd |
⊢ ( 𝜑 → ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ∈ ( 𝑌 𝐼 𝑋 ) ) |
23 |
20 22
|
sseldd |
⊢ ( 𝜑 → ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
24 |
1 16 17 8 11 4 5 6 19 23
|
issect2 |
⊢ ( 𝜑 → ( 𝐹 ( 𝑋 ( Sect ‘ 𝐶 ) 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ↔ ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ) ) |
25 |
9
|
a1i |
⊢ ( 𝜑 → ⚬ = ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) ) |
26 |
25
|
eqcomd |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) = ⚬ ) |
27 |
26
|
oveqd |
⊢ ( 𝜑 → ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ⚬ 𝐹 ) ) |
28 |
27
|
eqeq1d |
⊢ ( 𝜑 → ( ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐶 ) 𝑋 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ↔ ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ⚬ 𝐹 ) = ( 1 ‘ 𝑋 ) ) ) |
29 |
24 28
|
bitrd |
⊢ ( 𝜑 → ( 𝐹 ( 𝑋 ( Sect ‘ 𝐶 ) 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ↔ ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ⚬ 𝐹 ) = ( 1 ‘ 𝑋 ) ) ) |
30 |
15 29
|
mpbid |
⊢ ( 𝜑 → ( ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ⚬ 𝐹 ) = ( 1 ‘ 𝑋 ) ) |