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 |
|
eqid |
⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 ) |
10 |
|
eqid |
⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 ) |
11 |
1 2 3 4 5 6 7 8 9 10
|
curf1 |
⊢ ( 𝜑 → 𝐾 = 〈 ( 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 ) ) ) 〉 ) |
12 |
6
|
fvexi |
⊢ 𝐵 ∈ V |
13 |
12
|
mptex |
⊢ ( 𝑦 ∈ 𝐵 ↦ ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) ∈ V |
14 |
12 12
|
mpoex |
⊢ ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) ∈ V |
15 |
13 14
|
op1std |
⊢ ( 𝐾 = 〈 ( 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 ) ) ) 〉 → ( 1st ‘ 𝐾 ) = ( 𝑦 ∈ 𝐵 ↦ ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) ) |
16 |
11 15
|
syl |
⊢ ( 𝜑 → ( 1st ‘ 𝐾 ) = ( 𝑦 ∈ 𝐵 ↦ ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) ) |
17 |
13 14
|
op2ndd |
⊢ ( 𝐾 = 〈 ( 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 ) ) ) 〉 → ( 2nd ‘ 𝐾 ) = ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) ) |
18 |
11 17
|
syl |
⊢ ( 𝜑 → ( 2nd ‘ 𝐾 ) = ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) ) |
19 |
16 18
|
opeq12d |
⊢ ( 𝜑 → 〈 ( 1st ‘ 𝐾 ) , ( 2nd ‘ 𝐾 ) 〉 = 〈 ( 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 ) ) ) 〉 ) |
20 |
11 19
|
eqtr4d |
⊢ ( 𝜑 → 𝐾 = 〈 ( 1st ‘ 𝐾 ) , ( 2nd ‘ 𝐾 ) 〉 ) |
21 |
|
eqid |
⊢ ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 ) |
22 |
|
eqid |
⊢ ( Hom ‘ 𝐸 ) = ( Hom ‘ 𝐸 ) |
23 |
|
eqid |
⊢ ( Id ‘ 𝐷 ) = ( Id ‘ 𝐷 ) |
24 |
|
eqid |
⊢ ( Id ‘ 𝐸 ) = ( Id ‘ 𝐸 ) |
25 |
|
eqid |
⊢ ( comp ‘ 𝐷 ) = ( comp ‘ 𝐷 ) |
26 |
|
eqid |
⊢ ( comp ‘ 𝐸 ) = ( comp ‘ 𝐸 ) |
27 |
|
funcrcl |
⊢ ( 𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) → ( ( 𝐶 ×c 𝐷 ) ∈ Cat ∧ 𝐸 ∈ Cat ) ) |
28 |
5 27
|
syl |
⊢ ( 𝜑 → ( ( 𝐶 ×c 𝐷 ) ∈ Cat ∧ 𝐸 ∈ Cat ) ) |
29 |
28
|
simprd |
⊢ ( 𝜑 → 𝐸 ∈ Cat ) |
30 |
|
eqid |
⊢ ( 𝐶 ×c 𝐷 ) = ( 𝐶 ×c 𝐷 ) |
31 |
30 2 6
|
xpcbas |
⊢ ( 𝐴 × 𝐵 ) = ( Base ‘ ( 𝐶 ×c 𝐷 ) ) |
32 |
|
relfunc |
⊢ Rel ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) |
33 |
|
1st2ndbr |
⊢ ( ( Rel ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ∧ 𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ) → ( 1st ‘ 𝐹 ) ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ( 2nd ‘ 𝐹 ) ) |
34 |
32 5 33
|
sylancr |
⊢ ( 𝜑 → ( 1st ‘ 𝐹 ) ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ( 2nd ‘ 𝐹 ) ) |
35 |
31 21 34
|
funcf1 |
⊢ ( 𝜑 → ( 1st ‘ 𝐹 ) : ( 𝐴 × 𝐵 ) ⟶ ( Base ‘ 𝐸 ) ) |
36 |
35
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( 1st ‘ 𝐹 ) : ( 𝐴 × 𝐵 ) ⟶ ( Base ‘ 𝐸 ) ) |
37 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝑋 ∈ 𝐴 ) |
38 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
39 |
36 37 38
|
fovrnd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ∈ ( Base ‘ 𝐸 ) ) |
40 |
16 39
|
fmpt3d |
⊢ ( 𝜑 → ( 1st ‘ 𝐾 ) : 𝐵 ⟶ ( Base ‘ 𝐸 ) ) |
41 |
|
eqid |
⊢ ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) = ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) |
42 |
|
ovex |
⊢ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∈ V |
43 |
42
|
mptex |
⊢ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ∈ V |
44 |
41 43
|
fnmpoi |
⊢ ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) Fn ( 𝐵 × 𝐵 ) |
45 |
18
|
fneq1d |
⊢ ( 𝜑 → ( ( 2nd ‘ 𝐾 ) Fn ( 𝐵 × 𝐵 ) ↔ ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) Fn ( 𝐵 × 𝐵 ) ) ) |
46 |
44 45
|
mpbiri |
⊢ ( 𝜑 → ( 2nd ‘ 𝐾 ) Fn ( 𝐵 × 𝐵 ) ) |
47 |
18
|
oveqd |
⊢ ( 𝜑 → ( 𝑦 ( 2nd ‘ 𝐾 ) 𝑧 ) = ( 𝑦 ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) 𝑧 ) ) |
48 |
41
|
ovmpt4g |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ∈ V ) → ( 𝑦 ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) 𝑧 ) = ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) |
49 |
43 48
|
mp3an3 |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑦 ( 𝑦 ∈ 𝐵 , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) 𝑧 ) = ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) |
50 |
47 49
|
sylan9eq |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑦 ( 2nd ‘ 𝐾 ) 𝑧 ) = ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↦ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) ) |
51 |
|
eqid |
⊢ ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) = ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) |
52 |
34
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 1st ‘ 𝐹 ) ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ( 2nd ‘ 𝐹 ) ) |
53 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝑋 ∈ 𝐴 ) |
54 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝑦 ∈ 𝐵 ) |
55 |
53 54
|
opelxpd |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 〈 𝑋 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ) |
56 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝑧 ∈ 𝐵 ) |
57 |
53 56
|
opelxpd |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 〈 𝑋 , 𝑧 〉 ∈ ( 𝐴 × 𝐵 ) ) |
58 |
31 51 22 52 55 57
|
funcf2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) : ( 〈 𝑋 , 𝑦 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑧 〉 ) ⟶ ( ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐸 ) ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑧 〉 ) ) ) |
59 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
60 |
30 31 59 9 51 55 57
|
xpchom |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 〈 𝑋 , 𝑦 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑧 〉 ) = ( ( ( 1st ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐶 ) ( 1st ‘ 〈 𝑋 , 𝑧 〉 ) ) × ( ( 2nd ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐷 ) ( 2nd ‘ 〈 𝑋 , 𝑧 〉 ) ) ) ) |
61 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝐶 ∈ Cat ) |
62 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝐷 ∈ Cat ) |
63 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ) |
64 |
1 2 61 62 63 6 53 8 54
|
curf11 |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) = ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) |
65 |
|
df-ov |
⊢ ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) = ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) |
66 |
64 65
|
eqtr2di |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) = ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) ) |
67 |
1 2 61 62 63 6 53 8 56
|
curf11 |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) = ( 𝑋 ( 1st ‘ 𝐹 ) 𝑧 ) ) |
68 |
|
df-ov |
⊢ ( 𝑋 ( 1st ‘ 𝐹 ) 𝑧 ) = ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑧 〉 ) |
69 |
67 68
|
eqtr2di |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑧 〉 ) = ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) ) |
70 |
66 69
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐸 ) ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑧 〉 ) ) = ( ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) ( Hom ‘ 𝐸 ) ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) ) ) |
71 |
60 70
|
feq23d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) : ( 〈 𝑋 , 𝑦 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑧 〉 ) ⟶ ( ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐸 ) ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑧 〉 ) ) ↔ ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) : ( ( ( 1st ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐶 ) ( 1st ‘ 〈 𝑋 , 𝑧 〉 ) ) × ( ( 2nd ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐷 ) ( 2nd ‘ 〈 𝑋 , 𝑧 〉 ) ) ) ⟶ ( ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) ( Hom ‘ 𝐸 ) ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) ) ) ) |
72 |
58 71
|
mpbid |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) : ( ( ( 1st ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐶 ) ( 1st ‘ 〈 𝑋 , 𝑧 〉 ) ) × ( ( 2nd ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐷 ) ( 2nd ‘ 〈 𝑋 , 𝑧 〉 ) ) ) ⟶ ( ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) ( Hom ‘ 𝐸 ) ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) ) ) |
73 |
2 59 10 61 53
|
catidcl |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
74 |
|
op1stg |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 1st ‘ 〈 𝑋 , 𝑦 〉 ) = 𝑋 ) |
75 |
53 54 74
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 1st ‘ 〈 𝑋 , 𝑦 〉 ) = 𝑋 ) |
76 |
|
op1stg |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) → ( 1st ‘ 〈 𝑋 , 𝑧 〉 ) = 𝑋 ) |
77 |
53 56 76
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 1st ‘ 〈 𝑋 , 𝑧 〉 ) = 𝑋 ) |
78 |
75 77
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 1st ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐶 ) ( 1st ‘ 〈 𝑋 , 𝑧 〉 ) ) = ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
79 |
73 78
|
eleqtrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ∈ ( ( 1st ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐶 ) ( 1st ‘ 〈 𝑋 , 𝑧 〉 ) ) ) |
80 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) |
81 |
|
op2ndg |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ( 2nd ‘ 〈 𝑋 , 𝑦 〉 ) = 𝑦 ) |
82 |
53 54 81
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 2nd ‘ 〈 𝑋 , 𝑦 〉 ) = 𝑦 ) |
83 |
|
op2ndg |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) → ( 2nd ‘ 〈 𝑋 , 𝑧 〉 ) = 𝑧 ) |
84 |
53 56 83
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( 2nd ‘ 〈 𝑋 , 𝑧 〉 ) = 𝑧 ) |
85 |
82 84
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( 2nd ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐷 ) ( 2nd ‘ 〈 𝑋 , 𝑧 〉 ) ) = ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) |
86 |
80 85
|
eleqtrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → 𝑔 ∈ ( ( 2nd ‘ 〈 𝑋 , 𝑦 〉 ) ( Hom ‘ 𝐷 ) ( 2nd ‘ 〈 𝑋 , 𝑧 〉 ) ) ) |
87 |
72 79 86
|
fovrnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) → ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ∈ ( ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) ( Hom ‘ 𝐸 ) ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) ) ) |
88 |
50 87
|
fmpt3d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑦 ( 2nd ‘ 𝐾 ) 𝑧 ) : ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ⟶ ( ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) ( Hom ‘ 𝐸 ) ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) ) ) |
89 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝐶 ∈ Cat ) |
90 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝐷 ∈ Cat ) |
91 |
|
eqid |
⊢ ( Id ‘ ( 𝐶 ×c 𝐷 ) ) = ( Id ‘ ( 𝐶 ×c 𝐷 ) ) |
92 |
30 89 90 2 6 10 23 91 37 38
|
xpcid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( Id ‘ ( 𝐶 ×c 𝐷 ) ) ‘ 〈 𝑋 , 𝑦 〉 ) = 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) 〉 ) |
93 |
92
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑦 〉 ) ‘ ( ( Id ‘ ( 𝐶 ×c 𝐷 ) ) ‘ 〈 𝑋 , 𝑦 〉 ) ) = ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑦 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) 〉 ) ) |
94 |
|
df-ov |
⊢ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑦 〉 ) ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ) = ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑦 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) 〉 ) |
95 |
93 94
|
eqtr4di |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑦 〉 ) ‘ ( ( Id ‘ ( 𝐶 ×c 𝐷 ) ) ‘ 〈 𝑋 , 𝑦 〉 ) ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑦 〉 ) ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ) ) |
96 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( 1st ‘ 𝐹 ) ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ( 2nd ‘ 𝐹 ) ) |
97 |
|
opelxpi |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 〈 𝑋 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ) |
98 |
7 97
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 〈 𝑋 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ) |
99 |
31 91 24 96 98
|
funcid |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑦 〉 ) ‘ ( ( Id ‘ ( 𝐶 ×c 𝐷 ) ) ‘ 〈 𝑋 , 𝑦 〉 ) ) = ( ( Id ‘ 𝐸 ) ‘ ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) ) ) |
100 |
95 99
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑦 〉 ) ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ) = ( ( Id ‘ 𝐸 ) ‘ ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) ) ) |
101 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ) |
102 |
6 9 23 90 38
|
catidcl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑦 ) ) |
103 |
1 2 89 90 101 6 37 8 38 9 10 38 102
|
curf12 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑦 ( 2nd ‘ 𝐾 ) 𝑦 ) ‘ ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑦 〉 ) ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ) ) |
104 |
1 2 89 90 101 6 37 8 38
|
curf11 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) = ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) |
105 |
104 65
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) = ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) ) |
106 |
105
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( Id ‘ 𝐸 ) ‘ ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) ) = ( ( Id ‘ 𝐸 ) ‘ ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) ) ) |
107 |
100 103 106
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑦 ( 2nd ‘ 𝐾 ) 𝑦 ) ‘ ( ( Id ‘ 𝐷 ) ‘ 𝑦 ) ) = ( ( Id ‘ 𝐸 ) ‘ ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) ) ) |
108 |
7
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝑋 ∈ 𝐴 ) |
109 |
|
simp21 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝑦 ∈ 𝐵 ) |
110 |
|
simp22 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝑧 ∈ 𝐵 ) |
111 |
|
eqid |
⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) |
112 |
|
eqid |
⊢ ( comp ‘ ( 𝐶 ×c 𝐷 ) ) = ( comp ‘ ( 𝐶 ×c 𝐷 ) ) |
113 |
|
simp23 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝑤 ∈ 𝐵 ) |
114 |
3
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝐶 ∈ Cat ) |
115 |
2 59 10 114 108
|
catidcl |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) ) |
116 |
|
simp3l |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) |
117 |
|
simp3r |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) |
118 |
30 2 6 59 9 108 109 108 110 111 25 112 108 113 115 116 115 117
|
xpcco2 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ℎ 〉 ( 〈 〈 𝑋 , 𝑦 〉 , 〈 𝑋 , 𝑧 〉 〉 ( comp ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑤 〉 ) 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 〉 ) = 〈 ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑋 〉 ( comp ‘ 𝐶 ) 𝑋 ) ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) , ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) 〉 ) |
119 |
2 59 10 114 108 111 108 115
|
catlid |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑋 〉 ( comp ‘ 𝐶 ) 𝑋 ) ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) = ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) |
120 |
119
|
opeq1d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 〈 ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑋 〉 ( comp ‘ 𝐶 ) 𝑋 ) ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ) , ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) 〉 = 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) 〉 ) |
121 |
118 120
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ℎ 〉 ( 〈 〈 𝑋 , 𝑦 〉 , 〈 𝑋 , 𝑧 〉 〉 ( comp ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑤 〉 ) 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 〉 ) = 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) 〉 ) |
122 |
121
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ‘ ( 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ℎ 〉 ( 〈 〈 𝑋 , 𝑦 〉 , 〈 𝑋 , 𝑧 〉 〉 ( comp ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑤 〉 ) 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 〉 ) ) = ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) 〉 ) ) |
123 |
|
df-ov |
⊢ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ) = ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) 〉 ) |
124 |
122 123
|
eqtr4di |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ‘ ( 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ℎ 〉 ( 〈 〈 𝑋 , 𝑦 〉 , 〈 𝑋 , 𝑧 〉 〉 ( comp ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑤 〉 ) 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 〉 ) ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ) ) |
125 |
34
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( 1st ‘ 𝐹 ) ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ( 2nd ‘ 𝐹 ) ) |
126 |
108 109
|
opelxpd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 〈 𝑋 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ) |
127 |
108 110
|
opelxpd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 〈 𝑋 , 𝑧 〉 ∈ ( 𝐴 × 𝐵 ) ) |
128 |
108 113
|
opelxpd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 〈 𝑋 , 𝑤 〉 ∈ ( 𝐴 × 𝐵 ) ) |
129 |
115 116
|
opelxpd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 〉 ∈ ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) |
130 |
30 2 6 59 9 108 109 108 110 51
|
xpchom2 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( 〈 𝑋 , 𝑦 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑧 〉 ) = ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) |
131 |
129 130
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 〉 ∈ ( 〈 𝑋 , 𝑦 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑧 〉 ) ) |
132 |
115 117
|
opelxpd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ℎ 〉 ∈ ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) |
133 |
30 2 6 59 9 108 110 108 113 51
|
xpchom2 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( 〈 𝑋 , 𝑧 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑤 〉 ) = ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) |
134 |
132 133
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ℎ 〉 ∈ ( 〈 𝑋 , 𝑧 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑤 〉 ) ) |
135 |
31 51 112 26 125 126 127 128 131 134
|
funcco |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ‘ ( 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ℎ 〉 ( 〈 〈 𝑋 , 𝑦 〉 , 〈 𝑋 , 𝑧 〉 〉 ( comp ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑋 , 𝑤 〉 ) 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 〉 ) ) = ( ( ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ℎ 〉 ) ( 〈 ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) , ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑧 〉 ) 〉 ( comp ‘ 𝐸 ) ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑤 〉 ) ) ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 〉 ) ) ) |
136 |
124 135
|
eqtr3d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ) = ( ( ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ℎ 〉 ) ( 〈 ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) , ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑧 〉 ) 〉 ( comp ‘ 𝐸 ) ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑤 〉 ) ) ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 〉 ) ) ) |
137 |
4
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝐷 ∈ Cat ) |
138 |
5
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 𝐹 ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐸 ) ) |
139 |
6 9 25 137 109 110 113 116 117
|
catcocl |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑤 ) ) |
140 |
1 2 114 137 138 6 108 8 109 9 10 113 139
|
curf12 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 𝑦 ( 2nd ‘ 𝐾 ) 𝑤 ) ‘ ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ) ) |
141 |
1 2 114 137 138 6 108 8 109
|
curf11 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) = ( 𝑋 ( 1st ‘ 𝐹 ) 𝑦 ) ) |
142 |
141 65
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) = ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) ) |
143 |
1 2 114 137 138 6 108 8 110
|
curf11 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) = ( 𝑋 ( 1st ‘ 𝐹 ) 𝑧 ) ) |
144 |
143 68
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) = ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑧 〉 ) ) |
145 |
142 144
|
opeq12d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → 〈 ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) , ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) 〉 = 〈 ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) , ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑧 〉 ) 〉 ) |
146 |
1 2 114 137 138 6 108 8 113
|
curf11 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 1st ‘ 𝐾 ) ‘ 𝑤 ) = ( 𝑋 ( 1st ‘ 𝐹 ) 𝑤 ) ) |
147 |
|
df-ov |
⊢ ( 𝑋 ( 1st ‘ 𝐹 ) 𝑤 ) = ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑤 〉 ) |
148 |
146 147
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 1st ‘ 𝐾 ) ‘ 𝑤 ) = ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑤 〉 ) ) |
149 |
145 148
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( 〈 ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) , ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) 〉 ( comp ‘ 𝐸 ) ( ( 1st ‘ 𝐾 ) ‘ 𝑤 ) ) = ( 〈 ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) , ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑧 〉 ) 〉 ( comp ‘ 𝐸 ) ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑤 〉 ) ) ) |
150 |
1 2 114 137 138 6 108 8 110 9 10 113 117
|
curf12 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 𝑧 ( 2nd ‘ 𝐾 ) 𝑤 ) ‘ ℎ ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ℎ ) ) |
151 |
|
df-ov |
⊢ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ℎ ) = ( ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ℎ 〉 ) |
152 |
150 151
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 𝑧 ( 2nd ‘ 𝐾 ) 𝑤 ) ‘ ℎ ) = ( ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ℎ 〉 ) ) |
153 |
1 2 114 137 138 6 108 8 109 9 10 110 116
|
curf12 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 𝑦 ( 2nd ‘ 𝐾 ) 𝑧 ) ‘ 𝑔 ) = ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) ) |
154 |
|
df-ov |
⊢ ( ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) 𝑔 ) = ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 〉 ) |
155 |
153 154
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 𝑦 ( 2nd ‘ 𝐾 ) 𝑧 ) ‘ 𝑔 ) = ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 〉 ) ) |
156 |
149 152 155
|
oveq123d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( ( 𝑧 ( 2nd ‘ 𝐾 ) 𝑤 ) ‘ ℎ ) ( 〈 ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) , ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) 〉 ( comp ‘ 𝐸 ) ( ( 1st ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( 𝑦 ( 2nd ‘ 𝐾 ) 𝑧 ) ‘ 𝑔 ) ) = ( ( ( 〈 𝑋 , 𝑧 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑤 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , ℎ 〉 ) ( 〈 ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑦 〉 ) , ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑧 〉 ) 〉 ( comp ‘ 𝐸 ) ( ( 1st ‘ 𝐹 ) ‘ 〈 𝑋 , 𝑤 〉 ) ) ( ( 〈 𝑋 , 𝑦 〉 ( 2nd ‘ 𝐹 ) 〈 𝑋 , 𝑧 〉 ) ‘ 〈 ( ( Id ‘ 𝐶 ) ‘ 𝑋 ) , 𝑔 〉 ) ) ) |
157 |
136 140 156
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ) ) → ( ( 𝑦 ( 2nd ‘ 𝐾 ) 𝑤 ) ‘ ( ℎ ( 〈 𝑦 , 𝑧 〉 ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ) = ( ( ( 𝑧 ( 2nd ‘ 𝐾 ) 𝑤 ) ‘ ℎ ) ( 〈 ( ( 1st ‘ 𝐾 ) ‘ 𝑦 ) , ( ( 1st ‘ 𝐾 ) ‘ 𝑧 ) 〉 ( comp ‘ 𝐸 ) ( ( 1st ‘ 𝐾 ) ‘ 𝑤 ) ) ( ( 𝑦 ( 2nd ‘ 𝐾 ) 𝑧 ) ‘ 𝑔 ) ) ) |
158 |
6 21 9 22 23 24 25 26 4 29 40 46 88 107 157
|
isfuncd |
⊢ ( 𝜑 → ( 1st ‘ 𝐾 ) ( 𝐷 Func 𝐸 ) ( 2nd ‘ 𝐾 ) ) |
159 |
|
df-br |
⊢ ( ( 1st ‘ 𝐾 ) ( 𝐷 Func 𝐸 ) ( 2nd ‘ 𝐾 ) ↔ 〈 ( 1st ‘ 𝐾 ) , ( 2nd ‘ 𝐾 ) 〉 ∈ ( 𝐷 Func 𝐸 ) ) |
160 |
158 159
|
sylib |
⊢ ( 𝜑 → 〈 ( 1st ‘ 𝐾 ) , ( 2nd ‘ 𝐾 ) 〉 ∈ ( 𝐷 Func 𝐸 ) ) |
161 |
20 160
|
eqeltrd |
⊢ ( 𝜑 → 𝐾 ∈ ( 𝐷 Func 𝐸 ) ) |