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 |
1 2 3
|
diagval |
⊢ ( 𝜑 → 𝐿 = ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) |
10 |
9
|
fveq2d |
⊢ ( 𝜑 → ( 1st ‘ 𝐿 ) = ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ) |
11 |
10
|
fveq1d |
⊢ ( 𝜑 → ( ( 1st ‘ 𝐿 ) ‘ 𝑋 ) = ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) ) |
12 |
6 11
|
eqtrid |
⊢ ( 𝜑 → 𝐾 = ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) ) |
13 |
12
|
fveq2d |
⊢ ( 𝜑 → ( 1st ‘ 𝐾 ) = ( 1st ‘ ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) ) ) |
14 |
13
|
fveq1d |
⊢ ( 𝜑 → ( ( 1st ‘ 𝐾 ) ‘ 𝑌 ) = ( ( 1st ‘ ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) ) ‘ 𝑌 ) ) |
15 |
|
eqid |
⊢ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) = ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) |
16 |
|
eqid |
⊢ ( 𝐶 ×c 𝐷 ) = ( 𝐶 ×c 𝐷 ) |
17 |
|
eqid |
⊢ ( 𝐶 1stF 𝐷 ) = ( 𝐶 1stF 𝐷 ) |
18 |
16 2 3 17
|
1stfcl |
⊢ ( 𝜑 → ( 𝐶 1stF 𝐷 ) ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐶 ) ) |
19 |
|
eqid |
⊢ ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) = ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) |
20 |
15 4 2 3 18 7 5 19 8
|
curf11 |
⊢ ( 𝜑 → ( ( 1st ‘ ( ( 1st ‘ ( 〈 𝐶 , 𝐷 〉 curryF ( 𝐶 1stF 𝐷 ) ) ) ‘ 𝑋 ) ) ‘ 𝑌 ) = ( 𝑋 ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) 𝑌 ) ) |
21 |
|
df-ov |
⊢ ( 𝑋 ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) 𝑌 ) = ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) |
22 |
16 4 7
|
xpcbas |
⊢ ( 𝐴 × 𝐵 ) = ( Base ‘ ( 𝐶 ×c 𝐷 ) ) |
23 |
|
eqid |
⊢ ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) = ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) |
24 |
5 8
|
opelxpd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ( 𝐴 × 𝐵 ) ) |
25 |
16 22 23 2 3 17 24
|
1stf1 |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) = ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) ) |
26 |
21 25
|
eqtrid |
⊢ ( 𝜑 → ( 𝑋 ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) 𝑌 ) = ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) ) |
27 |
|
op1stg |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) → ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑋 ) |
28 |
5 8 27
|
syl2anc |
⊢ ( 𝜑 → ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑋 ) |
29 |
26 28
|
eqtrd |
⊢ ( 𝜑 → ( 𝑋 ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) 𝑌 ) = 𝑋 ) |
30 |
14 20 29
|
3eqtrd |
⊢ ( 𝜑 → ( ( 1st ‘ 𝐾 ) ‘ 𝑌 ) = 𝑋 ) |