Step |
Hyp |
Ref |
Expression |
1 |
|
hoi2toco.1 |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
hoi2toco.c |
⊢ 𝐼 = ( 𝑘 ∈ 𝑋 ↦ 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ) |
3 |
2
|
funmpt2 |
⊢ Fun 𝐼 |
4 |
3
|
a1i |
⊢ ( 𝜑 → Fun 𝐼 ) |
5 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → Fun 𝐼 ) |
6 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝑘 ∈ 𝑋 ) |
7 |
2
|
dmeqi |
⊢ dom 𝐼 = dom ( 𝑘 ∈ 𝑋 ↦ 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ) |
8 |
7
|
a1i |
⊢ ( 𝜑 → dom 𝐼 = dom ( 𝑘 ∈ 𝑋 ↦ 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ) ) |
9 |
|
opex |
⊢ 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ∈ V |
10 |
9
|
2a1i |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑋 → 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ∈ V ) ) |
11 |
1 10
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝑋 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ∈ V ) |
12 |
|
dmmptg |
⊢ ( ∀ 𝑘 ∈ 𝑋 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ∈ V → dom ( 𝑘 ∈ 𝑋 ↦ 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ) = 𝑋 ) |
13 |
11 12
|
syl |
⊢ ( 𝜑 → dom ( 𝑘 ∈ 𝑋 ↦ 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ) = 𝑋 ) |
14 |
8 13
|
eqtr2d |
⊢ ( 𝜑 → 𝑋 = dom 𝐼 ) |
15 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝑋 = dom 𝐼 ) |
16 |
6 15
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝑘 ∈ dom 𝐼 ) |
17 |
|
fvco |
⊢ ( ( Fun 𝐼 ∧ 𝑘 ∈ dom 𝐼 ) → ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = ( [,) ‘ ( 𝐼 ‘ 𝑘 ) ) ) |
18 |
5 16 17
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = ( [,) ‘ ( 𝐼 ‘ 𝑘 ) ) ) |
19 |
9
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ∈ V ) |
20 |
2
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝑋 ∧ 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ∈ V ) → ( 𝐼 ‘ 𝑘 ) = 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ) |
21 |
6 19 20
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐼 ‘ 𝑘 ) = 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ) |
22 |
21
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( [,) ‘ ( 𝐼 ‘ 𝑘 ) ) = ( [,) ‘ 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ) ) |
23 |
|
df-ov |
⊢ ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = ( [,) ‘ 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ) |
24 |
23
|
eqcomi |
⊢ ( [,) ‘ 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) |
25 |
24
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( [,) ‘ 〈 ( 𝐴 ‘ 𝑘 ) , ( 𝐵 ‘ 𝑘 ) 〉 ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
26 |
18 22 25
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
27 |
1 26
|
ixpeq2d |
⊢ ( 𝜑 → X 𝑘 ∈ 𝑋 ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = X 𝑘 ∈ 𝑋 ( ( 𝐴 ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |