Step |
Hyp |
Ref |
Expression |
1 |
|
curfpropd.1 |
⊢ ( 𝜑 → ( Homf ‘ 𝐴 ) = ( Homf ‘ 𝐵 ) ) |
2 |
|
curfpropd.2 |
⊢ ( 𝜑 → ( compf ‘ 𝐴 ) = ( compf ‘ 𝐵 ) ) |
3 |
|
curfpropd.3 |
⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) ) |
4 |
|
curfpropd.4 |
⊢ ( 𝜑 → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) ) |
5 |
|
curfpropd.a |
⊢ ( 𝜑 → 𝐴 ∈ Cat ) |
6 |
|
curfpropd.b |
⊢ ( 𝜑 → 𝐵 ∈ Cat ) |
7 |
|
curfpropd.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
8 |
|
curfpropd.d |
⊢ ( 𝜑 → 𝐷 ∈ Cat ) |
9 |
|
curfpropd.f |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐴 ×c 𝐶 ) Func 𝐸 ) ) |
10 |
1
|
homfeqbas |
⊢ ( 𝜑 → ( Base ‘ 𝐴 ) = ( Base ‘ 𝐵 ) ) |
11 |
3
|
homfeqbas |
⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) ) |
12 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) ) |
13 |
12
|
mpteq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) → ( 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) = ( 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) ) |
14 |
12
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
16 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
17 |
|
eqid |
⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 ) |
18 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) ∧ ( 𝑦 ∈ ( Base ‘ 𝐶 ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) ) |
19 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) ∧ ( 𝑦 ∈ ( Base ‘ 𝐶 ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) ) |
20 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) ∧ ( 𝑦 ∈ ( Base ‘ 𝐶 ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑧 ∈ ( Base ‘ 𝐶 ) ) |
21 |
15 16 17 18 19 20
|
homfeqval |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) ∧ ( 𝑦 ∈ ( Base ‘ 𝐶 ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) |
22 |
1 2 5 6
|
cidpropd |
⊢ ( 𝜑 → ( Id ‘ 𝐴 ) = ( Id ‘ 𝐵 ) ) |
23 |
22
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) ∧ ( 𝑦 ∈ ( Base ‘ 𝐶 ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ) → ( Id ‘ 𝐴 ) = ( Id ‘ 𝐵 ) ) |
24 |
23
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) ∧ ( 𝑦 ∈ ( Base ‘ 𝐶 ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( Id ‘ 𝐴 ) ‘ 𝑥 ) = ( ( Id ‘ 𝐵 ) ‘ 𝑥 ) ) |
25 |
24
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) ∧ ( 𝑦 ∈ ( Base ‘ 𝐶 ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( ( Id ‘ 𝐴 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) = ( ( ( Id ‘ 𝐵 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) |
26 |
21 25
|
mpteq12dv |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) ∧ ( 𝑦 ∈ ( Base ‘ 𝐶 ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐴 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) = ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐵 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) |
27 |
12 14 26
|
mpoeq123dva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) → ( 𝑦 ∈ ( Base ‘ 𝐶 ) , 𝑧 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐴 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) = ( 𝑦 ∈ ( Base ‘ 𝐷 ) , 𝑧 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐵 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) ) |
28 |
13 27
|
opeq12d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) → 〈 ( 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ ( Base ‘ 𝐶 ) , 𝑧 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐴 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 = 〈 ( 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ ( Base ‘ 𝐷 ) , 𝑧 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐵 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) |
29 |
10 28
|
mpteq12dva |
⊢ ( 𝜑 → ( 𝑥 ∈ ( Base ‘ 𝐴 ) ↦ 〈 ( 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ ( Base ‘ 𝐶 ) , 𝑧 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐴 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) = ( 𝑥 ∈ ( Base ‘ 𝐵 ) ↦ 〈 ( 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ ( Base ‘ 𝐷 ) , 𝑧 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐵 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) ) |
30 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐴 ) ) → ( Base ‘ 𝐴 ) = ( Base ‘ 𝐵 ) ) |
31 |
|
eqid |
⊢ ( Base ‘ 𝐴 ) = ( Base ‘ 𝐴 ) |
32 |
|
eqid |
⊢ ( Hom ‘ 𝐴 ) = ( Hom ‘ 𝐴 ) |
33 |
|
eqid |
⊢ ( Hom ‘ 𝐵 ) = ( Hom ‘ 𝐵 ) |
34 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐴 ) ∧ 𝑦 ∈ ( Base ‘ 𝐴 ) ) ) → ( Homf ‘ 𝐴 ) = ( Homf ‘ 𝐵 ) ) |
35 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐴 ) ∧ 𝑦 ∈ ( Base ‘ 𝐴 ) ) ) → 𝑥 ∈ ( Base ‘ 𝐴 ) ) |
36 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐴 ) ∧ 𝑦 ∈ ( Base ‘ 𝐴 ) ) ) → 𝑦 ∈ ( Base ‘ 𝐴 ) ) |
37 |
31 32 33 34 35 36
|
homfeqval |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐴 ) ∧ 𝑦 ∈ ( Base ‘ 𝐴 ) ) ) → ( 𝑥 ( Hom ‘ 𝐴 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐵 ) 𝑦 ) ) |
38 |
11
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐴 ) ∧ 𝑦 ∈ ( Base ‘ 𝐴 ) ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐴 ) 𝑦 ) ) → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) ) |
39 |
3 4 7 8
|
cidpropd |
⊢ ( 𝜑 → ( Id ‘ 𝐶 ) = ( Id ‘ 𝐷 ) ) |
40 |
39
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐴 ) ∧ 𝑦 ∈ ( Base ‘ 𝐴 ) ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐴 ) 𝑦 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( Id ‘ 𝐶 ) = ( Id ‘ 𝐷 ) ) |
41 |
40
|
fveq1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐴 ) ∧ 𝑦 ∈ ( Base ‘ 𝐴 ) ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐴 ) 𝑦 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( ( Id ‘ 𝐶 ) ‘ 𝑧 ) = ( ( Id ‘ 𝐷 ) ‘ 𝑧 ) ) |
42 |
41
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐴 ) ∧ 𝑦 ∈ ( Base ‘ 𝐴 ) ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐴 ) 𝑦 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( ( Id ‘ 𝐶 ) ‘ 𝑧 ) ) = ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( ( Id ‘ 𝐷 ) ‘ 𝑧 ) ) ) |
43 |
38 42
|
mpteq12dva |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐴 ) ∧ 𝑦 ∈ ( Base ‘ 𝐴 ) ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐴 ) 𝑦 ) ) → ( 𝑧 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( ( Id ‘ 𝐶 ) ‘ 𝑧 ) ) ) = ( 𝑧 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( ( Id ‘ 𝐷 ) ‘ 𝑧 ) ) ) ) |
44 |
37 43
|
mpteq12dva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐴 ) ∧ 𝑦 ∈ ( Base ‘ 𝐴 ) ) ) → ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐴 ) 𝑦 ) ↦ ( 𝑧 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( ( Id ‘ 𝐶 ) ‘ 𝑧 ) ) ) ) = ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐵 ) 𝑦 ) ↦ ( 𝑧 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( ( Id ‘ 𝐷 ) ‘ 𝑧 ) ) ) ) ) |
45 |
10 30 44
|
mpoeq123dva |
⊢ ( 𝜑 → ( 𝑥 ∈ ( Base ‘ 𝐴 ) , 𝑦 ∈ ( Base ‘ 𝐴 ) ↦ ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐴 ) 𝑦 ) ↦ ( 𝑧 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( ( Id ‘ 𝐶 ) ‘ 𝑧 ) ) ) ) ) = ( 𝑥 ∈ ( Base ‘ 𝐵 ) , 𝑦 ∈ ( Base ‘ 𝐵 ) ↦ ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐵 ) 𝑦 ) ↦ ( 𝑧 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( ( Id ‘ 𝐷 ) ‘ 𝑧 ) ) ) ) ) ) |
46 |
29 45
|
opeq12d |
⊢ ( 𝜑 → 〈 ( 𝑥 ∈ ( Base ‘ 𝐴 ) ↦ 〈 ( 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ ( Base ‘ 𝐶 ) , 𝑧 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐴 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) , ( 𝑥 ∈ ( Base ‘ 𝐴 ) , 𝑦 ∈ ( Base ‘ 𝐴 ) ↦ ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐴 ) 𝑦 ) ↦ ( 𝑧 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( ( Id ‘ 𝐶 ) ‘ 𝑧 ) ) ) ) ) 〉 = 〈 ( 𝑥 ∈ ( Base ‘ 𝐵 ) ↦ 〈 ( 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ ( Base ‘ 𝐷 ) , 𝑧 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐵 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) , ( 𝑥 ∈ ( Base ‘ 𝐵 ) , 𝑦 ∈ ( Base ‘ 𝐵 ) ↦ ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐵 ) 𝑦 ) ↦ ( 𝑧 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( ( Id ‘ 𝐷 ) ‘ 𝑧 ) ) ) ) ) 〉 ) |
47 |
|
eqid |
⊢ ( 〈 𝐴 , 𝐶 〉 curryF 𝐹 ) = ( 〈 𝐴 , 𝐶 〉 curryF 𝐹 ) |
48 |
|
eqid |
⊢ ( Id ‘ 𝐴 ) = ( Id ‘ 𝐴 ) |
49 |
|
eqid |
⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 ) |
50 |
47 31 5 7 9 15 16 48 32 49
|
curfval |
⊢ ( 𝜑 → ( 〈 𝐴 , 𝐶 〉 curryF 𝐹 ) = 〈 ( 𝑥 ∈ ( Base ‘ 𝐴 ) ↦ 〈 ( 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ ( Base ‘ 𝐶 ) , 𝑧 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐴 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) , ( 𝑥 ∈ ( Base ‘ 𝐴 ) , 𝑦 ∈ ( Base ‘ 𝐴 ) ↦ ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐴 ) 𝑦 ) ↦ ( 𝑧 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( ( Id ‘ 𝐶 ) ‘ 𝑧 ) ) ) ) ) 〉 ) |
51 |
|
eqid |
⊢ ( 〈 𝐵 , 𝐷 〉 curryF 𝐹 ) = ( 〈 𝐵 , 𝐷 〉 curryF 𝐹 ) |
52 |
|
eqid |
⊢ ( Base ‘ 𝐵 ) = ( Base ‘ 𝐵 ) |
53 |
1 2 3 4 5 6 7 8
|
xpcpropd |
⊢ ( 𝜑 → ( 𝐴 ×c 𝐶 ) = ( 𝐵 ×c 𝐷 ) ) |
54 |
53
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐴 ×c 𝐶 ) Func 𝐸 ) = ( ( 𝐵 ×c 𝐷 ) Func 𝐸 ) ) |
55 |
9 54
|
eleqtrd |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐵 ×c 𝐷 ) Func 𝐸 ) ) |
56 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
57 |
|
eqid |
⊢ ( Id ‘ 𝐵 ) = ( Id ‘ 𝐵 ) |
58 |
|
eqid |
⊢ ( Id ‘ 𝐷 ) = ( Id ‘ 𝐷 ) |
59 |
51 52 6 8 55 56 17 57 33 58
|
curfval |
⊢ ( 𝜑 → ( 〈 𝐵 , 𝐷 〉 curryF 𝐹 ) = 〈 ( 𝑥 ∈ ( Base ‘ 𝐵 ) ↦ 〈 ( 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ ( Base ‘ 𝐷 ) , 𝑧 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐵 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) , ( 𝑥 ∈ ( Base ‘ 𝐵 ) , 𝑦 ∈ ( Base ‘ 𝐵 ) ↦ ( 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐵 ) 𝑦 ) ↦ ( 𝑧 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( ( Id ‘ 𝐷 ) ‘ 𝑧 ) ) ) ) ) 〉 ) |
60 |
46 50 59
|
3eqtr4d |
⊢ ( 𝜑 → ( 〈 𝐴 , 𝐶 〉 curryF 𝐹 ) = ( 〈 𝐵 , 𝐷 〉 curryF 𝐹 ) ) |