Step |
Hyp |
Ref |
Expression |
1 |
|
prfval.k |
⊢ 𝑃 = ( 𝐹 〈,〉F 𝐺 ) |
2 |
|
prfval.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
3 |
|
prfval.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
4 |
|
prfval.c |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 Func 𝐷 ) ) |
5 |
|
prfval.d |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝐶 Func 𝐸 ) ) |
6 |
|
df-prf |
⊢ 〈,〉F = ( 𝑓 ∈ V , 𝑔 ∈ V ↦ ⦋ dom ( 1st ‘ 𝑓 ) / 𝑏 ⦌ 〈 ( 𝑥 ∈ 𝑏 ↦ 〈 ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝑔 ) ‘ 𝑥 ) 〉 ) , ( 𝑥 ∈ 𝑏 , 𝑦 ∈ 𝑏 ↦ ( ℎ ∈ dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 ) ) 〉 ) |
7 |
6
|
a1i |
⊢ ( 𝜑 → 〈,〉F = ( 𝑓 ∈ V , 𝑔 ∈ V ↦ ⦋ dom ( 1st ‘ 𝑓 ) / 𝑏 ⦌ 〈 ( 𝑥 ∈ 𝑏 ↦ 〈 ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝑔 ) ‘ 𝑥 ) 〉 ) , ( 𝑥 ∈ 𝑏 , 𝑦 ∈ 𝑏 ↦ ( ℎ ∈ dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 ) ) 〉 ) ) |
8 |
|
fvex |
⊢ ( 1st ‘ 𝑓 ) ∈ V |
9 |
8
|
dmex |
⊢ dom ( 1st ‘ 𝑓 ) ∈ V |
10 |
9
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → dom ( 1st ‘ 𝑓 ) ∈ V ) |
11 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → 𝑓 = 𝐹 ) |
12 |
11
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → ( 1st ‘ 𝑓 ) = ( 1st ‘ 𝐹 ) ) |
13 |
12
|
dmeqd |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → dom ( 1st ‘ 𝑓 ) = dom ( 1st ‘ 𝐹 ) ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
15 |
|
relfunc |
⊢ Rel ( 𝐶 Func 𝐷 ) |
16 |
|
1st2ndbr |
⊢ ( ( Rel ( 𝐶 Func 𝐷 ) ∧ 𝐹 ∈ ( 𝐶 Func 𝐷 ) ) → ( 1st ‘ 𝐹 ) ( 𝐶 Func 𝐷 ) ( 2nd ‘ 𝐹 ) ) |
17 |
15 4 16
|
sylancr |
⊢ ( 𝜑 → ( 1st ‘ 𝐹 ) ( 𝐶 Func 𝐷 ) ( 2nd ‘ 𝐹 ) ) |
18 |
2 14 17
|
funcf1 |
⊢ ( 𝜑 → ( 1st ‘ 𝐹 ) : 𝐵 ⟶ ( Base ‘ 𝐷 ) ) |
19 |
18
|
fdmd |
⊢ ( 𝜑 → dom ( 1st ‘ 𝐹 ) = 𝐵 ) |
20 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → dom ( 1st ‘ 𝐹 ) = 𝐵 ) |
21 |
13 20
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → dom ( 1st ‘ 𝑓 ) = 𝐵 ) |
22 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → 𝑏 = 𝐵 ) |
23 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → 𝑓 = 𝐹 ) |
24 |
23
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → ( 1st ‘ 𝑓 ) = ( 1st ‘ 𝐹 ) ) |
25 |
24
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) = ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) ) |
26 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → 𝑔 = 𝐺 ) |
27 |
26
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → ( 1st ‘ 𝑔 ) = ( 1st ‘ 𝐺 ) ) |
28 |
27
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → ( ( 1st ‘ 𝑔 ) ‘ 𝑥 ) = ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) ) |
29 |
25 28
|
opeq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → 〈 ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝑔 ) ‘ 𝑥 ) 〉 = 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) |
30 |
22 29
|
mpteq12dv |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → ( 𝑥 ∈ 𝑏 ↦ 〈 ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝑔 ) ‘ 𝑥 ) 〉 ) = ( 𝑥 ∈ 𝐵 ↦ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) ) |
31 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → ( ℎ ∈ dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 ) = ( ℎ ∈ dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 ) ) |
32 |
22 22 31
|
mpoeq123dv |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → ( 𝑥 ∈ 𝑏 , 𝑦 ∈ 𝑏 ↦ ( ℎ ∈ dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( ℎ ∈ dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 ) ) ) |
33 |
23
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → 𝑓 = 𝐹 ) |
34 |
33
|
fveq2d |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( 2nd ‘ 𝑓 ) = ( 2nd ‘ 𝐹 ) ) |
35 |
34
|
oveqd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) = ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ) |
36 |
35
|
dmeqd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) = dom ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ) |
37 |
|
eqid |
⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 ) |
38 |
17
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( 1st ‘ 𝐹 ) ( 𝐶 Func 𝐷 ) ( 2nd ‘ 𝐹 ) ) |
39 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
40 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
41 |
2 3 37 38 39 40
|
funcf2 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( ( 1st ‘ 𝐹 ) ‘ 𝑦 ) ) ) |
42 |
41
|
fdmd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → dom ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) = ( 𝑥 𝐻 𝑦 ) ) |
43 |
36 42
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) = ( 𝑥 𝐻 𝑦 ) ) |
44 |
35
|
fveq1d |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) = ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) ) |
45 |
26
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → 𝑔 = 𝐺 ) |
46 |
45
|
fveq2d |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( 2nd ‘ 𝑔 ) = ( 2nd ‘ 𝐺 ) ) |
47 |
46
|
oveqd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) = ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ) |
48 |
47
|
fveq1d |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) = ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) ) |
49 |
44 48
|
opeq12d |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 = 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) |
50 |
43 49
|
mpteq12dv |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐵 ) → ( ℎ ∈ dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 ) = ( ℎ ∈ ( 𝑥 𝐻 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) |
51 |
50
|
3impa |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ℎ ∈ dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 ) = ( ℎ ∈ ( 𝑥 𝐻 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) |
52 |
51
|
mpoeq3dva |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( ℎ ∈ dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( ℎ ∈ ( 𝑥 𝐻 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) ) |
53 |
32 52
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → ( 𝑥 ∈ 𝑏 , 𝑦 ∈ 𝑏 ↦ ( ℎ ∈ dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( ℎ ∈ ( 𝑥 𝐻 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) ) |
54 |
30 53
|
opeq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) ∧ 𝑏 = 𝐵 ) → 〈 ( 𝑥 ∈ 𝑏 ↦ 〈 ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝑔 ) ‘ 𝑥 ) 〉 ) , ( 𝑥 ∈ 𝑏 , 𝑦 ∈ 𝑏 ↦ ( ℎ ∈ dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 ) ) 〉 = 〈 ( 𝑥 ∈ 𝐵 ↦ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( ℎ ∈ ( 𝑥 𝐻 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) 〉 ) |
55 |
10 21 54
|
csbied2 |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → ⦋ dom ( 1st ‘ 𝑓 ) / 𝑏 ⦌ 〈 ( 𝑥 ∈ 𝑏 ↦ 〈 ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝑔 ) ‘ 𝑥 ) 〉 ) , ( 𝑥 ∈ 𝑏 , 𝑦 ∈ 𝑏 ↦ ( ℎ ∈ dom ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) 〉 ) ) 〉 = 〈 ( 𝑥 ∈ 𝐵 ↦ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( ℎ ∈ ( 𝑥 𝐻 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) 〉 ) |
56 |
4
|
elexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
57 |
5
|
elexd |
⊢ ( 𝜑 → 𝐺 ∈ V ) |
58 |
|
opex |
⊢ 〈 ( 𝑥 ∈ 𝐵 ↦ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( ℎ ∈ ( 𝑥 𝐻 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) 〉 ∈ V |
59 |
58
|
a1i |
⊢ ( 𝜑 → 〈 ( 𝑥 ∈ 𝐵 ↦ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( ℎ ∈ ( 𝑥 𝐻 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) 〉 ∈ V ) |
60 |
7 55 56 57 59
|
ovmpod |
⊢ ( 𝜑 → ( 𝐹 〈,〉F 𝐺 ) = 〈 ( 𝑥 ∈ 𝐵 ↦ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( ℎ ∈ ( 𝑥 𝐻 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) 〉 ) |
61 |
1 60
|
eqtrid |
⊢ ( 𝜑 → 𝑃 = 〈 ( 𝑥 ∈ 𝐵 ↦ 〈 ( ( 1st ‘ 𝐹 ) ‘ 𝑥 ) , ( ( 1st ‘ 𝐺 ) ‘ 𝑥 ) 〉 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( ℎ ∈ ( 𝑥 𝐻 𝑦 ) ↦ 〈 ( ( 𝑥 ( 2nd ‘ 𝐹 ) 𝑦 ) ‘ ℎ ) , ( ( 𝑥 ( 2nd ‘ 𝐺 ) 𝑦 ) ‘ ℎ ) 〉 ) ) 〉 ) |