Step |
Hyp |
Ref |
Expression |
1 |
|
curfval.g |
⊢ 𝐺 = ( 〈 𝐶 , 𝐷 〉 curryF 𝐹 ) |
2 |
|
curfval.a |
⊢ 𝐴 = ( Base ‘ 𝐶 ) |
3 |
|
curfval.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
4 |
|
curfval.d |
⊢ ( 𝜑 → 𝐷 ∈ Cat ) |
5 |
|
curfval.f |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ) |
6 |
|
curfval.b |
⊢ 𝐵 = ( Base ‘ 𝐷 ) |
7 |
|
curf1.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐴 ) |
8 |
|
curf1.k |
⊢ 𝐾 = ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) |
9 |
|
curf1.j |
⊢ 𝐽 = ( Hom ‘ 𝐷 ) |
10 |
|
curf1.1 |
⊢ 1 = ( Id ‘ 𝐶 ) |
11 |
1 2 3 4 5 6 9 10
|
curf1fval |
⊢ ( 𝜑 → ( 1st ‘ 𝐺 ) = ( 𝑥 ∈ 𝐴 ↦ 〈 ( 𝑦 ∈ 𝐵 ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ↦ ( ( 1 ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) ) |
12 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → 𝑥 = 𝑋 ) |
13 |
12
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) = ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) |
14 |
13
|
mpteq2dv |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝑦 ∈ 𝐵 ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) = ( 𝑦 ∈ 𝐵 ↦ ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) ) |
15 |
|
simp1r |
⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → 𝑥 = 𝑋 ) |
16 |
15
|
opeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → 〈 𝑥 , 𝑦 〉 = 〈 𝑋 , 𝑦 〉 ) |
17 |
15
|
opeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → 〈 𝑥 , 𝑧 〉 = 〈 𝑋 , 𝑧 〉 ) |
18 |
16 17
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) = ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) ) |
19 |
15
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 1 ‘ 𝑥 ) = ( 1 ‘ 𝑋 ) ) |
20 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → 𝑔 = 𝑔 ) |
21 |
18 19 20
|
oveq123d |
⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 1 ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) = ( ( 1 ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) |
22 |
21
|
mpteq2dv |
⊢ ( ( ( 𝜑 ∧ 𝑥 = 𝑋 ) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ↦ ( ( 1 ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) = ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ↦ ( ( 1 ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) |
23 |
22
|
mpoeq3dva |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ↦ ( ( 1 ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) = ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ↦ ( ( 1 ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) ) |
24 |
14 23
|
opeq12d |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → 〈 ( 𝑦 ∈ 𝐵 ↦ ( 𝑥 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ↦ ( ( 1 ‘ 𝑥 ) ( 〈 𝑥 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑥 , 𝑧 〉 ) 𝑔 ) ) ) 〉 = 〈 ( 𝑦 ∈ 𝐵 ↦ ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ↦ ( ( 1 ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) |
25 |
|
opex |
⊢ 〈 ( 𝑦 ∈ 𝐵 ↦ ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ↦ ( ( 1 ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ∈ V |
26 |
25
|
a1i |
⊢ ( 𝜑 → 〈 ( 𝑦 ∈ 𝐵 ↦ ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ↦ ( ( 1 ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ∈ V ) |
27 |
11 24 7 26
|
fvmptd |
⊢ ( 𝜑 → ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) = 〈 ( 𝑦 ∈ 𝐵 ↦ ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ↦ ( ( 1 ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) |
28 |
8 27
|
syl5eq |
⊢ ( 𝜑 → 𝐾 = 〈 ( 𝑦 ∈ 𝐵 ↦ ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) , ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ↦ ( ( 1 ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) 〉 ) |