Step |
Hyp |
Ref |
Expression |
1 |
|
funcsect.b |
⊢ 𝐵 = ( Base ‘ 𝐷 ) |
2 |
|
funcsect.s |
⊢ 𝑆 = ( Sect ‘ 𝐷 ) |
3 |
|
funcsect.t |
⊢ 𝑇 = ( Sect ‘ 𝐸 ) |
4 |
|
funcsect.f |
⊢ ( 𝜑 → 𝐹 ( 𝐷 Func 𝐸 ) 𝐺 ) |
5 |
|
funcsect.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
funcsect.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
|
funcsect.m |
⊢ ( 𝜑 → 𝑀 ( 𝑋 𝑆 𝑌 ) 𝑁 ) |
8 |
|
eqid |
⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 ) |
9 |
|
eqid |
⊢ ( comp ‘ 𝐷 ) = ( comp ‘ 𝐷 ) |
10 |
|
eqid |
⊢ ( Id ‘ 𝐷 ) = ( Id ‘ 𝐷 ) |
11 |
|
df-br |
⊢ ( 𝐹 ( 𝐷 Func 𝐸 ) 𝐺 ↔ 〈 𝐹 , 𝐺 〉 ∈ ( 𝐷 Func 𝐸 ) ) |
12 |
4 11
|
sylib |
⊢ ( 𝜑 → 〈 𝐹 , 𝐺 〉 ∈ ( 𝐷 Func 𝐸 ) ) |
13 |
|
funcrcl |
⊢ ( 〈 𝐹 , 𝐺 〉 ∈ ( 𝐷 Func 𝐸 ) → ( 𝐷 ∈ Cat ∧ 𝐸 ∈ Cat ) ) |
14 |
12 13
|
syl |
⊢ ( 𝜑 → ( 𝐷 ∈ Cat ∧ 𝐸 ∈ Cat ) ) |
15 |
14
|
simpld |
⊢ ( 𝜑 → 𝐷 ∈ Cat ) |
16 |
1 8 9 10 2 15 5 6
|
issect |
⊢ ( 𝜑 → ( 𝑀 ( 𝑋 𝑆 𝑌 ) 𝑁 ↔ ( 𝑀 ∈ ( 𝑋 ( Hom ‘ 𝐷 ) 𝑌 ) ∧ 𝑁 ∈ ( 𝑌 ( Hom ‘ 𝐷 ) 𝑋 ) ∧ ( 𝑁 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐷 ) 𝑋 ) 𝑀 ) = ( ( Id ‘ 𝐷 ) ‘ 𝑋 ) ) ) ) |
17 |
7 16
|
mpbid |
⊢ ( 𝜑 → ( 𝑀 ∈ ( 𝑋 ( Hom ‘ 𝐷 ) 𝑌 ) ∧ 𝑁 ∈ ( 𝑌 ( Hom ‘ 𝐷 ) 𝑋 ) ∧ ( 𝑁 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐷 ) 𝑋 ) 𝑀 ) = ( ( Id ‘ 𝐷 ) ‘ 𝑋 ) ) ) |
18 |
17
|
simp3d |
⊢ ( 𝜑 → ( 𝑁 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐷 ) 𝑋 ) 𝑀 ) = ( ( Id ‘ 𝐷 ) ‘ 𝑋 ) ) |
19 |
18
|
fveq2d |
⊢ ( 𝜑 → ( ( 𝑋 𝐺 𝑋 ) ‘ ( 𝑁 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐷 ) 𝑋 ) 𝑀 ) ) = ( ( 𝑋 𝐺 𝑋 ) ‘ ( ( Id ‘ 𝐷 ) ‘ 𝑋 ) ) ) |
20 |
|
eqid |
⊢ ( comp ‘ 𝐸 ) = ( comp ‘ 𝐸 ) |
21 |
17
|
simp1d |
⊢ ( 𝜑 → 𝑀 ∈ ( 𝑋 ( Hom ‘ 𝐷 ) 𝑌 ) ) |
22 |
17
|
simp2d |
⊢ ( 𝜑 → 𝑁 ∈ ( 𝑌 ( Hom ‘ 𝐷 ) 𝑋 ) ) |
23 |
1 8 9 20 4 5 6 5 21 22
|
funcco |
⊢ ( 𝜑 → ( ( 𝑋 𝐺 𝑋 ) ‘ ( 𝑁 ( 〈 𝑋 , 𝑌 〉 ( comp ‘ 𝐷 ) 𝑋 ) 𝑀 ) ) = ( ( ( 𝑌 𝐺 𝑋 ) ‘ 𝑁 ) ( 〈 ( 𝐹 ‘ 𝑋 ) , ( 𝐹 ‘ 𝑌 ) 〉 ( comp ‘ 𝐸 ) ( 𝐹 ‘ 𝑋 ) ) ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑀 ) ) ) |
24 |
|
eqid |
⊢ ( Id ‘ 𝐸 ) = ( Id ‘ 𝐸 ) |
25 |
1 10 24 4 5
|
funcid |
⊢ ( 𝜑 → ( ( 𝑋 𝐺 𝑋 ) ‘ ( ( Id ‘ 𝐷 ) ‘ 𝑋 ) ) = ( ( Id ‘ 𝐸 ) ‘ ( 𝐹 ‘ 𝑋 ) ) ) |
26 |
19 23 25
|
3eqtr3d |
⊢ ( 𝜑 → ( ( ( 𝑌 𝐺 𝑋 ) ‘ 𝑁 ) ( 〈 ( 𝐹 ‘ 𝑋 ) , ( 𝐹 ‘ 𝑌 ) 〉 ( comp ‘ 𝐸 ) ( 𝐹 ‘ 𝑋 ) ) ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑀 ) ) = ( ( Id ‘ 𝐸 ) ‘ ( 𝐹 ‘ 𝑋 ) ) ) |
27 |
|
eqid |
⊢ ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 ) |
28 |
|
eqid |
⊢ ( Hom ‘ 𝐸 ) = ( Hom ‘ 𝐸 ) |
29 |
14
|
simprd |
⊢ ( 𝜑 → 𝐸 ∈ Cat ) |
30 |
1 27 4
|
funcf1 |
⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ ( Base ‘ 𝐸 ) ) |
31 |
30 5
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ∈ ( Base ‘ 𝐸 ) ) |
32 |
30 6
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑌 ) ∈ ( Base ‘ 𝐸 ) ) |
33 |
1 8 28 4 5 6
|
funcf2 |
⊢ ( 𝜑 → ( 𝑋 𝐺 𝑌 ) : ( 𝑋 ( Hom ‘ 𝐷 ) 𝑌 ) ⟶ ( ( 𝐹 ‘ 𝑋 ) ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑌 ) ) ) |
34 |
33 21
|
ffvelrnd |
⊢ ( 𝜑 → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑀 ) ∈ ( ( 𝐹 ‘ 𝑋 ) ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑌 ) ) ) |
35 |
1 8 28 4 6 5
|
funcf2 |
⊢ ( 𝜑 → ( 𝑌 𝐺 𝑋 ) : ( 𝑌 ( Hom ‘ 𝐷 ) 𝑋 ) ⟶ ( ( 𝐹 ‘ 𝑌 ) ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑋 ) ) ) |
36 |
35 22
|
ffvelrnd |
⊢ ( 𝜑 → ( ( 𝑌 𝐺 𝑋 ) ‘ 𝑁 ) ∈ ( ( 𝐹 ‘ 𝑌 ) ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑋 ) ) ) |
37 |
27 28 20 24 3 29 31 32 34 36
|
issect2 |
⊢ ( 𝜑 → ( ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑀 ) ( ( 𝐹 ‘ 𝑋 ) 𝑇 ( 𝐹 ‘ 𝑌 ) ) ( ( 𝑌 𝐺 𝑋 ) ‘ 𝑁 ) ↔ ( ( ( 𝑌 𝐺 𝑋 ) ‘ 𝑁 ) ( 〈 ( 𝐹 ‘ 𝑋 ) , ( 𝐹 ‘ 𝑌 ) 〉 ( comp ‘ 𝐸 ) ( 𝐹 ‘ 𝑋 ) ) ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑀 ) ) = ( ( Id ‘ 𝐸 ) ‘ ( 𝐹 ‘ 𝑋 ) ) ) ) |
38 |
26 37
|
mpbird |
⊢ ( 𝜑 → ( ( 𝑋 𝐺 𝑌 ) ‘ 𝑀 ) ( ( 𝐹 ‘ 𝑋 ) 𝑇 ( 𝐹 ‘ 𝑌 ) ) ( ( 𝑌 𝐺 𝑋 ) ‘ 𝑁 ) ) |