Step |
Hyp |
Ref |
Expression |
1 |
|
curf2.g |
⊢ 𝐺 = ( 〈 𝐶 , 𝐷 〉 curryF 𝐹 ) |
2 |
|
curf2.a |
⊢ 𝐴 = ( Base ‘ 𝐶 ) |
3 |
|
curf2.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
4 |
|
curf2.d |
⊢ ( 𝜑 → 𝐷 ∈ Cat ) |
5 |
|
curf2.f |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ) |
6 |
|
curf2.b |
⊢ 𝐵 = ( Base ‘ 𝐷 ) |
7 |
|
curf2.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
8 |
|
curf2.i |
⊢ 𝐼 = ( Id ‘ 𝐷 ) |
9 |
|
curf2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐴 ) |
10 |
|
curf2.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐴 ) |
11 |
|
curf2.k |
⊢ ( 𝜑 → 𝐾 ∈ ( 𝑋 𝐻 𝑌 ) ) |
12 |
|
curf2.l |
⊢ 𝐿 = ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑌 ) ‘ 𝐾 ) |
13 |
|
eqid |
⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 ) |
14 |
|
eqid |
⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 ) |
15 |
1 2 3 4 5 6 13 14 7 8
|
curfval |
⊢ ( 𝜑 → 𝐺 = 〈 ( 𝑥 ∈ 𝐴 ↦ 〈 ( 𝑦 ∈ 𝐵 ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) , ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐴 ↦ ( 𝑔 ∈ ( 𝑥 𝐻 𝑦 ) ↦ ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) ) 〉 ) |
16 |
2
|
fvexi |
⊢ 𝐴 ∈ V |
17 |
16
|
mptex |
⊢ ( 𝑥 ∈ 𝐴 ↦ 〈 ( 𝑦 ∈ 𝐵 ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) ∈ V |
18 |
16 16
|
mpoex |
⊢ ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐴 ↦ ( 𝑔 ∈ ( 𝑥 𝐻 𝑦 ) ↦ ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) ) ∈ V |
19 |
17 18
|
op2ndd |
⊢ ( 𝐺 = 〈 ( 𝑥 ∈ 𝐴 ↦ 〈 ( 𝑦 ∈ 𝐵 ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) , ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐴 ↦ ( 𝑔 ∈ ( 𝑥 𝐻 𝑦 ) ↦ ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) ) 〉 → ( 2nd ‘ 𝐺 ) = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐴 ↦ ( 𝑔 ∈ ( 𝑥 𝐻 𝑦 ) ↦ ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) ) ) |
20 |
15 19
|
syl |
⊢ ( 𝜑 → ( 2nd ‘ 𝐺 ) = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐴 ↦ ( 𝑔 ∈ ( 𝑥 𝐻 𝑦 ) ↦ ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) ) ) |
21 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → 𝑌 ∈ 𝐴 ) |
22 |
|
ovex |
⊢ ( 𝑥 𝐻 𝑦 ) ∈ V |
23 |
22
|
mptex |
⊢ ( 𝑔 ∈ ( 𝑥 𝐻 𝑦 ) ↦ ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) ∈ V |
24 |
23
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑔 ∈ ( 𝑥 𝐻 𝑦 ) ↦ ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) ∈ V ) |
25 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝐾 ∈ ( 𝑋 𝐻 𝑌 ) ) |
26 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝑥 = 𝑋 ) |
27 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝑦 = 𝑌 ) |
28 |
26 27
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑥 𝐻 𝑦 ) = ( 𝑋 𝐻 𝑌 ) ) |
29 |
25 28
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝐾 ∈ ( 𝑥 𝐻 𝑦 ) ) |
30 |
6
|
fvexi |
⊢ 𝐵 ∈ V |
31 |
30
|
mptex |
⊢ ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ∈ V |
32 |
31
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑔 = 𝐾 ) → ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ∈ V ) |
33 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑔 = 𝐾 ) → 𝑥 = 𝑋 ) |
34 |
33
|
opeq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑔 = 𝐾 ) → 〈 𝑥 , 𝑧 〉 = 〈 𝑋 , 𝑧 〉 ) |
35 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑔 = 𝐾 ) → 𝑦 = 𝑌 ) |
36 |
35
|
opeq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑔 = 𝐾 ) → 〈 𝑦 , 𝑧 〉 = 〈 𝑌 , 𝑧 〉 ) |
37 |
34 36
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑔 = 𝐾 ) → ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) = ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑌 , 𝑧 〉 ) ) |
38 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑔 = 𝐾 ) → 𝑔 = 𝐾 ) |
39 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑔 = 𝐾 ) → ( 𝐼 ‘ 𝑧 ) = ( 𝐼 ‘ 𝑧 ) ) |
40 |
37 38 39
|
oveq123d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑔 = 𝐾 ) → ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) = ( 𝐾 ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑌 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) |
41 |
40
|
mpteq2dv |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) ∧ 𝑔 = 𝐾 ) → ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) = ( 𝑧 ∈ 𝐵 ↦ ( 𝐾 ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑌 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) |
42 |
29 32 41
|
fvmptdv2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑌 ) = ( 𝑔 ∈ ( 𝑥 𝐻 𝑦 ) ↦ ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) → ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑌 ) ‘ 𝐾 ) = ( 𝑧 ∈ 𝐵 ↦ ( 𝐾 ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑌 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) ) |
43 |
9 21 24 42
|
ovmpodv |
⊢ ( 𝜑 → ( ( 2nd ‘ 𝐺 ) = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐴 ↦ ( 𝑔 ∈ ( 𝑥 𝐻 𝑦 ) ↦ ( 𝑧 ∈ 𝐵 ↦ ( 𝑔 ( 〈 𝑥 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑦 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) ) → ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑌 ) ‘ 𝐾 ) = ( 𝑧 ∈ 𝐵 ↦ ( 𝐾 ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑌 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) ) |
44 |
20 43
|
mpd |
⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑌 ) ‘ 𝐾 ) = ( 𝑧 ∈ 𝐵 ↦ ( 𝐾 ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑌 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) |
45 |
12 44
|
eqtrid |
⊢ ( 𝜑 → 𝐿 = ( 𝑧 ∈ 𝐵 ↦ ( 𝐾 ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑌 , 𝑧 〉 ) ( 𝐼 ‘ 𝑧 ) ) ) ) |