Step |
Hyp |
Ref |
Expression |
1 |
|
diagval.l |
⊢ 𝐿 = ( 𝐶 Δfunc 𝐷 ) |
2 |
|
diagval.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
3 |
|
diagval.d |
⊢ ( 𝜑 → 𝐷 ∈ Cat ) |
4 |
|
diag11.a |
⊢ 𝐴 = ( Base ‘ 𝐶 ) |
5 |
|
diag11.c |
⊢ ( 𝜑 → 𝑋 ∈ 𝐴 ) |
6 |
|
diag11.k |
⊢ 𝐾 = ( ( 1st ‘ 𝐿 ) ‘ 𝑋 ) |
7 |
|
diag11.b |
⊢ 𝐵 = ( Base ‘ 𝐷 ) |
8 |
|
diag11.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
9 |
|
diag12.j |
⊢ 𝐽 = ( Hom ‘ 𝐷 ) |
10 |
|
diag12.i |
⊢ 1 = ( Id ‘ 𝐶 ) |
11 |
|
diag12.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝐵 ) |
12 |
|
diag12.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑌 𝐽 𝑍 ) ) |
13 |
1 2 3
|
diagval |
⊢ ( 𝜑 → 𝐿 = ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) |
14 |
13
|
fveq2d |
⊢ ( 𝜑 → ( 1st ‘ 𝐿 ) = ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ) |
15 |
14
|
fveq1d |
⊢ ( 𝜑 → ( ( 1st ‘ 𝐿 ) ‘ 𝑋 ) = ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) ) |
16 |
6 15
|
eqtrid |
⊢ ( 𝜑 → 𝐾 = ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) ) |
17 |
16
|
fveq2d |
⊢ ( 𝜑 → ( 2nd ‘ 𝐾 ) = ( 2nd ‘ ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) ) ) |
18 |
17
|
oveqd |
⊢ ( 𝜑 → ( 𝑌 ( 2nd ‘ 𝐾 ) 𝑍 ) = ( 𝑌 ( 2nd ‘ ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) ) 𝑍 ) ) |
19 |
18
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝑌 ( 2nd ‘ 𝐾 ) 𝑍 ) ‘ 𝐹 ) = ( ( 𝑌 ( 2nd ‘ ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) ) 𝑍 ) ‘ 𝐹 ) ) |
20 |
|
eqid |
⊢ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) = ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) |
21 |
|
eqid |
⊢ ( 𝐶 ×c 𝐷 ) = ( 𝐶 ×c 𝐷 ) |
22 |
|
eqid |
⊢ ( 𝐶 1stF 𝐷 ) = ( 𝐶 1stF 𝐷 ) |
23 |
21 2 3 22
|
1stfcl |
⊢ ( 𝜑 → ( 𝐶 1stF 𝐷 ) ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐶 ) ) |
24 |
|
eqid |
⊢ ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) = ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) |
25 |
20 4 2 3 23 7 5 24 8 9 10 11 12
|
curf12 |
⊢ ( 𝜑 → ( ( 𝑌 ( 2nd ‘ ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) ) 𝑍 ) ‘ 𝐹 ) = ( ( 1 ‘ 𝑋 ) ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 1stF 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) 𝐹 ) ) |
26 |
|
df-ov |
⊢ ( ( 1 ‘ 𝑋 ) ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 1stF 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) 𝐹 ) = ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 1stF 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) ‘ 〈 ( 1 ‘ 𝑋 ) , 𝐹 〉 ) |
27 |
21 4 7
|
xpcbas |
⊢ ( 𝐴 × 𝐵 ) = ( Base ‘ ( 𝐶 ×c 𝐷 ) ) |
28 |
|
eqid |
⊢ ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) = ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) |
29 |
5 8
|
opelxpd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ( 𝐴 × 𝐵 ) ) |
30 |
5 11
|
opelxpd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑍 〉 ∈ ( 𝐴 × 𝐵 ) ) |
31 |
21 27 28 2 3 22 29 30
|
1stf2 |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 1stF 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) = ( 1st ↾ ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) ) ) |
32 |
31
|
fveq1d |
⊢ ( 𝜑 → ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 1stF 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) ‘ 〈 ( 1 ‘ 𝑋 ) , 𝐹 〉 ) = ( ( 1st ↾ ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) ) ‘ 〈 ( 1 ‘ 𝑋 ) , 𝐹 〉 ) ) |
33 |
26 32
|
eqtrid |
⊢ ( 𝜑 → ( ( 1 ‘ 𝑋 ) ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 1stF 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) 𝐹 ) = ( ( 1st ↾ ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) ) ‘ 〈 ( 1 ‘ 𝑋 ) , 𝐹 〉 ) ) |
34 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
35 |
4 34 10 2 5
|
catidcl |
⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
36 |
35 12
|
opelxpd |
⊢ ( 𝜑 → 〈 ( 1 ‘ 𝑋 ) , 𝐹 〉 ∈ ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑌 𝐽 𝑍 ) ) ) |
37 |
21 4 7 34 9 5 8 5 11 28
|
xpchom2 |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) = ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑌 𝐽 𝑍 ) ) ) |
38 |
36 37
|
eleqtrrd |
⊢ ( 𝜑 → 〈 ( 1 ‘ 𝑋 ) , 𝐹 〉 ∈ ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) ) |
39 |
38
|
fvresd |
⊢ ( 𝜑 → ( ( 1st ↾ ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) ) ‘ 〈 ( 1 ‘ 𝑋 ) , 𝐹 〉 ) = ( 1st ‘ 〈 ( 1 ‘ 𝑋 ) , 𝐹 〉 ) ) |
40 |
|
op1stg |
⊢ ( ( ( 1 ‘ 𝑋 ) ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) ∧ 𝐹 ∈ ( 𝑌 𝐽 𝑍 ) ) → ( 1st ‘ 〈 ( 1 ‘ 𝑋 ) , 𝐹 〉 ) = ( 1 ‘ 𝑋 ) ) |
41 |
35 12 40
|
syl2anc |
⊢ ( 𝜑 → ( 1st ‘ 〈 ( 1 ‘ 𝑋 ) , 𝐹 〉 ) = ( 1 ‘ 𝑋 ) ) |
42 |
33 39 41
|
3eqtrd |
⊢ ( 𝜑 → ( ( 1 ‘ 𝑋 ) ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 1stF 𝐷 ) ) 〈 𝑋 , 𝑍 〉 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ) |
43 |
19 25 42
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝑌 ( 2nd ‘ 𝐾 ) 𝑍 ) ‘ 𝐹 ) = ( 1 ‘ 𝑋 ) ) |