Step |
Hyp |
Ref |
Expression |
1 |
|
uncfval.g |
⊢ 𝐹 = ( 〈“ 𝐶 𝐷 𝐸 ”〉 uncurryF 𝐺 ) |
2 |
|
uncfval.c |
⊢ ( 𝜑 → 𝐷 ∈ Cat ) |
3 |
|
uncfval.d |
⊢ ( 𝜑 → 𝐸 ∈ Cat ) |
4 |
|
uncfval.f |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝐶 Func ( 𝐷 FuncCat 𝐸 ) ) ) |
5 |
|
uncf1.a |
⊢ 𝐴 = ( Base ‘ 𝐶 ) |
6 |
|
uncf1.b |
⊢ 𝐵 = ( Base ‘ 𝐷 ) |
7 |
|
uncf1.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐴 ) |
8 |
|
uncf1.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
9 |
|
uncf2.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
10 |
|
uncf2.j |
⊢ 𝐽 = ( Hom ‘ 𝐷 ) |
11 |
|
uncf2.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝐴 ) |
12 |
|
uncf2.w |
⊢ ( 𝜑 → 𝑊 ∈ 𝐵 ) |
13 |
|
uncf2.r |
⊢ ( 𝜑 → 𝑅 ∈ ( 𝑋 𝐻 𝑍 ) ) |
14 |
|
uncf2.s |
⊢ ( 𝜑 → 𝑆 ∈ ( 𝑌 𝐽 𝑊 ) ) |
15 |
1 2 3 4
|
uncfval |
⊢ ( 𝜑 → 𝐹 = ( ( 𝐷 evalF 𝐸 ) ∘func ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ) |
16 |
15
|
fveq2d |
⊢ ( 𝜑 → ( 2nd ‘ 𝐹 ) = ( 2nd ‘ ( ( 𝐷 evalF 𝐸 ) ∘func ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ) ) |
17 |
16
|
oveqd |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ 𝐹 ) 〈 𝑍 , 𝑊 〉 ) = ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐷 evalF 𝐸 ) ∘func ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ) 〈 𝑍 , 𝑊 〉 ) ) |
18 |
17
|
oveqd |
⊢ ( 𝜑 → ( 𝑅 ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ 𝐹 ) 〈 𝑍 , 𝑊 〉 ) 𝑆 ) = ( 𝑅 ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐷 evalF 𝐸 ) ∘func ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ) 〈 𝑍 , 𝑊 〉 ) 𝑆 ) ) |
19 |
|
df-ov |
⊢ ( 𝑅 ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐷 evalF 𝐸 ) ∘func ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ) 〈 𝑍 , 𝑊 〉 ) 𝑆 ) = ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐷 evalF 𝐸 ) ∘func ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) |
20 |
|
eqid |
⊢ ( 𝐶 ×c 𝐷 ) = ( 𝐶 ×c 𝐷 ) |
21 |
20 5 6
|
xpcbas |
⊢ ( 𝐴 × 𝐵 ) = ( Base ‘ ( 𝐶 ×c 𝐷 ) ) |
22 |
|
eqid |
⊢ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) = ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) |
23 |
|
eqid |
⊢ ( ( 𝐷 FuncCat 𝐸 ) ×c 𝐷 ) = ( ( 𝐷 FuncCat 𝐸 ) ×c 𝐷 ) |
24 |
|
funcrcl |
⊢ ( 𝐺 ∈ ( 𝐶 Func ( 𝐷 FuncCat 𝐸 ) ) → ( 𝐶 ∈ Cat ∧ ( 𝐷 FuncCat 𝐸 ) ∈ Cat ) ) |
25 |
4 24
|
syl |
⊢ ( 𝜑 → ( 𝐶 ∈ Cat ∧ ( 𝐷 FuncCat 𝐸 ) ∈ Cat ) ) |
26 |
25
|
simpld |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
27 |
|
eqid |
⊢ ( 𝐶 1stF 𝐷 ) = ( 𝐶 1stF 𝐷 ) |
28 |
20 26 2 27
|
1stfcl |
⊢ ( 𝜑 → ( 𝐶 1stF 𝐷 ) ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐶 ) ) |
29 |
28 4
|
cofucl |
⊢ ( 𝜑 → ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ∈ ( ( 𝐶 ×c 𝐷 ) Func ( 𝐷 FuncCat 𝐸 ) ) ) |
30 |
|
eqid |
⊢ ( 𝐶 2ndF 𝐷 ) = ( 𝐶 2ndF 𝐷 ) |
31 |
20 26 2 30
|
2ndfcl |
⊢ ( 𝜑 → ( 𝐶 2ndF 𝐷 ) ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐷 ) ) |
32 |
22 23 29 31
|
prfcl |
⊢ ( 𝜑 → ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ∈ ( ( 𝐶 ×c 𝐷 ) Func ( ( 𝐷 FuncCat 𝐸 ) ×c 𝐷 ) ) ) |
33 |
|
eqid |
⊢ ( 𝐷 evalF 𝐸 ) = ( 𝐷 evalF 𝐸 ) |
34 |
|
eqid |
⊢ ( 𝐷 FuncCat 𝐸 ) = ( 𝐷 FuncCat 𝐸 ) |
35 |
33 34 2 3
|
evlfcl |
⊢ ( 𝜑 → ( 𝐷 evalF 𝐸 ) ∈ ( ( ( 𝐷 FuncCat 𝐸 ) ×c 𝐷 ) Func 𝐸 ) ) |
36 |
7 8
|
opelxpd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ( 𝐴 × 𝐵 ) ) |
37 |
11 12
|
opelxpd |
⊢ ( 𝜑 → 〈 𝑍 , 𝑊 〉 ∈ ( 𝐴 × 𝐵 ) ) |
38 |
|
eqid |
⊢ ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) = ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) |
39 |
13 14
|
opelxpd |
⊢ ( 𝜑 → 〈 𝑅 , 𝑆 〉 ∈ ( ( 𝑋 𝐻 𝑍 ) × ( 𝑌 𝐽 𝑊 ) ) ) |
40 |
20 5 6 9 10 7 8 11 12 38
|
xpchom2 |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) = ( ( 𝑋 𝐻 𝑍 ) × ( 𝑌 𝐽 𝑊 ) ) ) |
41 |
39 40
|
eleqtrrd |
⊢ ( 𝜑 → 〈 𝑅 , 𝑆 〉 ∈ ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ) |
42 |
21 32 35 36 37 38 41
|
cofu2 |
⊢ ( 𝜑 → ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐷 evalF 𝐸 ) ∘func ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) = ( ( ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑍 , 𝑊 〉 ) ) ‘ ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) ) ) |
43 |
19 42
|
syl5eq |
⊢ ( 𝜑 → ( 𝑅 ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐷 evalF 𝐸 ) ∘func ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ) 〈 𝑍 , 𝑊 〉 ) 𝑆 ) = ( ( ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑍 , 𝑊 〉 ) ) ‘ ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) ) ) |
44 |
18 43
|
eqtrd |
⊢ ( 𝜑 → ( 𝑅 ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ 𝐹 ) 〈 𝑍 , 𝑊 〉 ) 𝑆 ) = ( ( ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑍 , 𝑊 〉 ) ) ‘ ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) ) ) |
45 |
22 21 38 29 31 36
|
prf1 |
⊢ ( 𝜑 → ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑋 , 𝑌 〉 ) = 〈 ( ( 1st ‘ ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ) ‘ 〈 𝑋 , 𝑌 〉 ) , ( ( 1st ‘ ( 𝐶 2ndF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) 〉 ) |
46 |
21 28 4 36
|
cofu1 |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ) ‘ 〈 𝑋 , 𝑌 〉 ) = ( ( 1st ‘ 𝐺 ) ‘ ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ) ) |
47 |
20 21 38 26 2 27 36
|
1stf1 |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) = ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) ) |
48 |
|
op1stg |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) → ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑋 ) |
49 |
7 8 48
|
syl2anc |
⊢ ( 𝜑 → ( 1st ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑋 ) |
50 |
47 49
|
eqtrd |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑋 ) |
51 |
50
|
fveq2d |
⊢ ( 𝜑 → ( ( 1st ‘ 𝐺 ) ‘ ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ) = ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) ) |
52 |
46 51
|
eqtrd |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ) ‘ 〈 𝑋 , 𝑌 〉 ) = ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) ) |
53 |
20 21 38 26 2 30 36
|
2ndf1 |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐶 2ndF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) = ( 2nd ‘ 〈 𝑋 , 𝑌 〉 ) ) |
54 |
|
op2ndg |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) → ( 2nd ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑌 ) |
55 |
7 8 54
|
syl2anc |
⊢ ( 𝜑 → ( 2nd ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑌 ) |
56 |
53 55
|
eqtrd |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐶 2ndF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) = 𝑌 ) |
57 |
52 56
|
opeq12d |
⊢ ( 𝜑 → 〈 ( ( 1st ‘ ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ) ‘ 〈 𝑋 , 𝑌 〉 ) , ( ( 1st ‘ ( 𝐶 2ndF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) 〉 = 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) , 𝑌 〉 ) |
58 |
45 57
|
eqtrd |
⊢ ( 𝜑 → ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑋 , 𝑌 〉 ) = 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) , 𝑌 〉 ) |
59 |
22 21 38 29 31 37
|
prf1 |
⊢ ( 𝜑 → ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑍 , 𝑊 〉 ) = 〈 ( ( 1st ‘ ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ) ‘ 〈 𝑍 , 𝑊 〉 ) , ( ( 1st ‘ ( 𝐶 2ndF 𝐷 ) ) ‘ 〈 𝑍 , 𝑊 〉 ) 〉 ) |
60 |
21 28 4 37
|
cofu1 |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ) ‘ 〈 𝑍 , 𝑊 〉 ) = ( ( 1st ‘ 𝐺 ) ‘ ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑍 , 𝑊 〉 ) ) ) |
61 |
20 21 38 26 2 27 37
|
1stf1 |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑍 , 𝑊 〉 ) = ( 1st ‘ 〈 𝑍 , 𝑊 〉 ) ) |
62 |
|
op1stg |
⊢ ( ( 𝑍 ∈ 𝐴 ∧ 𝑊 ∈ 𝐵 ) → ( 1st ‘ 〈 𝑍 , 𝑊 〉 ) = 𝑍 ) |
63 |
11 12 62
|
syl2anc |
⊢ ( 𝜑 → ( 1st ‘ 〈 𝑍 , 𝑊 〉 ) = 𝑍 ) |
64 |
61 63
|
eqtrd |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑍 , 𝑊 〉 ) = 𝑍 ) |
65 |
64
|
fveq2d |
⊢ ( 𝜑 → ( ( 1st ‘ 𝐺 ) ‘ ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑍 , 𝑊 〉 ) ) = ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) ) |
66 |
60 65
|
eqtrd |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ) ‘ 〈 𝑍 , 𝑊 〉 ) = ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) ) |
67 |
20 21 38 26 2 30 37
|
2ndf1 |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐶 2ndF 𝐷 ) ) ‘ 〈 𝑍 , 𝑊 〉 ) = ( 2nd ‘ 〈 𝑍 , 𝑊 〉 ) ) |
68 |
|
op2ndg |
⊢ ( ( 𝑍 ∈ 𝐴 ∧ 𝑊 ∈ 𝐵 ) → ( 2nd ‘ 〈 𝑍 , 𝑊 〉 ) = 𝑊 ) |
69 |
11 12 68
|
syl2anc |
⊢ ( 𝜑 → ( 2nd ‘ 〈 𝑍 , 𝑊 〉 ) = 𝑊 ) |
70 |
67 69
|
eqtrd |
⊢ ( 𝜑 → ( ( 1st ‘ ( 𝐶 2ndF 𝐷 ) ) ‘ 〈 𝑍 , 𝑊 〉 ) = 𝑊 ) |
71 |
66 70
|
opeq12d |
⊢ ( 𝜑 → 〈 ( ( 1st ‘ ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ) ‘ 〈 𝑍 , 𝑊 〉 ) , ( ( 1st ‘ ( 𝐶 2ndF 𝐷 ) ) ‘ 〈 𝑍 , 𝑊 〉 ) 〉 = 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) , 𝑊 〉 ) |
72 |
59 71
|
eqtrd |
⊢ ( 𝜑 → ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑍 , 𝑊 〉 ) = 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) , 𝑊 〉 ) |
73 |
58 72
|
oveq12d |
⊢ ( 𝜑 → ( ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑍 , 𝑊 〉 ) ) = ( 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) , 𝑌 〉 ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) , 𝑊 〉 ) ) |
74 |
22 21 38 29 31 36 37 41
|
prf2 |
⊢ ( 𝜑 → ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) = 〈 ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) , ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 2ndF 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) 〉 ) |
75 |
21 28 4 36 37 38 41
|
cofu2 |
⊢ ( 𝜑 → ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) = ( ( ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ( 2nd ‘ 𝐺 ) ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑍 , 𝑊 〉 ) ) ‘ ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 1stF 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) ) ) |
76 |
50 64
|
oveq12d |
⊢ ( 𝜑 → ( ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ( 2nd ‘ 𝐺 ) ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑍 , 𝑊 〉 ) ) = ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ) |
77 |
20 21 38 26 2 27 36 37
|
1stf2 |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 1stF 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) = ( 1st ↾ ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ) ) |
78 |
77
|
fveq1d |
⊢ ( 𝜑 → ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 1stF 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) = ( ( 1st ↾ ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ) ‘ 〈 𝑅 , 𝑆 〉 ) ) |
79 |
41
|
fvresd |
⊢ ( 𝜑 → ( ( 1st ↾ ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ) ‘ 〈 𝑅 , 𝑆 〉 ) = ( 1st ‘ 〈 𝑅 , 𝑆 〉 ) ) |
80 |
|
op1stg |
⊢ ( ( 𝑅 ∈ ( 𝑋 𝐻 𝑍 ) ∧ 𝑆 ∈ ( 𝑌 𝐽 𝑊 ) ) → ( 1st ‘ 〈 𝑅 , 𝑆 〉 ) = 𝑅 ) |
81 |
13 14 80
|
syl2anc |
⊢ ( 𝜑 → ( 1st ‘ 〈 𝑅 , 𝑆 〉 ) = 𝑅 ) |
82 |
78 79 81
|
3eqtrd |
⊢ ( 𝜑 → ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 1stF 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) = 𝑅 ) |
83 |
76 82
|
fveq12d |
⊢ ( 𝜑 → ( ( ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ( 2nd ‘ 𝐺 ) ( ( 1st ‘ ( 𝐶 1stF 𝐷 ) ) ‘ 〈 𝑍 , 𝑊 〉 ) ) ‘ ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 1stF 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) ) = ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ‘ 𝑅 ) ) |
84 |
75 83
|
eqtrd |
⊢ ( 𝜑 → ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) = ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ‘ 𝑅 ) ) |
85 |
20 21 38 26 2 30 36 37
|
2ndf2 |
⊢ ( 𝜑 → ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 2ndF 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) = ( 2nd ↾ ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ) ) |
86 |
85
|
fveq1d |
⊢ ( 𝜑 → ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 2ndF 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) = ( ( 2nd ↾ ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ) ‘ 〈 𝑅 , 𝑆 〉 ) ) |
87 |
41
|
fvresd |
⊢ ( 𝜑 → ( ( 2nd ↾ ( 〈 𝑋 , 𝑌 〉 ( Hom ‘ ( 𝐶 ×c 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ) ‘ 〈 𝑅 , 𝑆 〉 ) = ( 2nd ‘ 〈 𝑅 , 𝑆 〉 ) ) |
88 |
|
op2ndg |
⊢ ( ( 𝑅 ∈ ( 𝑋 𝐻 𝑍 ) ∧ 𝑆 ∈ ( 𝑌 𝐽 𝑊 ) ) → ( 2nd ‘ 〈 𝑅 , 𝑆 〉 ) = 𝑆 ) |
89 |
13 14 88
|
syl2anc |
⊢ ( 𝜑 → ( 2nd ‘ 〈 𝑅 , 𝑆 〉 ) = 𝑆 ) |
90 |
86 87 89
|
3eqtrd |
⊢ ( 𝜑 → ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 2ndF 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) = 𝑆 ) |
91 |
84 90
|
opeq12d |
⊢ ( 𝜑 → 〈 ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) , ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( 𝐶 2ndF 𝐷 ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) 〉 = 〈 ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ‘ 𝑅 ) , 𝑆 〉 ) |
92 |
74 91
|
eqtrd |
⊢ ( 𝜑 → ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) = 〈 ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ‘ 𝑅 ) , 𝑆 〉 ) |
93 |
73 92
|
fveq12d |
⊢ ( 𝜑 → ( ( ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑍 , 𝑊 〉 ) ) ‘ ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) ) = ( ( 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) , 𝑌 〉 ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) , 𝑊 〉 ) ‘ 〈 ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ‘ 𝑅 ) , 𝑆 〉 ) ) |
94 |
|
df-ov |
⊢ ( ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ‘ 𝑅 ) ( 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) , 𝑌 〉 ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) , 𝑊 〉 ) 𝑆 ) = ( ( 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) , 𝑌 〉 ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) , 𝑊 〉 ) ‘ 〈 ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ‘ 𝑅 ) , 𝑆 〉 ) |
95 |
93 94
|
eqtr4di |
⊢ ( 𝜑 → ( ( ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) ( ( 1st ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) ‘ 〈 𝑍 , 𝑊 〉 ) ) ‘ ( ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ ( ( 𝐺 ∘func ( 𝐶 1stF 𝐷 ) ) 〈,〉F ( 𝐶 2ndF 𝐷 ) ) ) 〈 𝑍 , 𝑊 〉 ) ‘ 〈 𝑅 , 𝑆 〉 ) ) = ( ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ‘ 𝑅 ) ( 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) , 𝑌 〉 ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) , 𝑊 〉 ) 𝑆 ) ) |
96 |
|
eqid |
⊢ ( comp ‘ 𝐸 ) = ( comp ‘ 𝐸 ) |
97 |
|
eqid |
⊢ ( 𝐷 Nat 𝐸 ) = ( 𝐷 Nat 𝐸 ) |
98 |
34
|
fucbas |
⊢ ( 𝐷 Func 𝐸 ) = ( Base ‘ ( 𝐷 FuncCat 𝐸 ) ) |
99 |
|
relfunc |
⊢ Rel ( 𝐶 Func ( 𝐷 FuncCat 𝐸 ) ) |
100 |
|
1st2ndbr |
⊢ ( ( Rel ( 𝐶 Func ( 𝐷 FuncCat 𝐸 ) ) ∧ 𝐺 ∈ ( 𝐶 Func ( 𝐷 FuncCat 𝐸 ) ) ) → ( 1st ‘ 𝐺 ) ( 𝐶 Func ( 𝐷 FuncCat 𝐸 ) ) ( 2nd ‘ 𝐺 ) ) |
101 |
99 4 100
|
sylancr |
⊢ ( 𝜑 → ( 1st ‘ 𝐺 ) ( 𝐶 Func ( 𝐷 FuncCat 𝐸 ) ) ( 2nd ‘ 𝐺 ) ) |
102 |
5 98 101
|
funcf1 |
⊢ ( 𝜑 → ( 1st ‘ 𝐺 ) : 𝐴 ⟶ ( 𝐷 Func 𝐸 ) ) |
103 |
102 7
|
ffvelrnd |
⊢ ( 𝜑 → ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) ∈ ( 𝐷 Func 𝐸 ) ) |
104 |
102 11
|
ffvelrnd |
⊢ ( 𝜑 → ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) ∈ ( 𝐷 Func 𝐸 ) ) |
105 |
|
eqid |
⊢ ( 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) , 𝑌 〉 ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) , 𝑊 〉 ) = ( 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) , 𝑌 〉 ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) , 𝑊 〉 ) |
106 |
34 97
|
fuchom |
⊢ ( 𝐷 Nat 𝐸 ) = ( Hom ‘ ( 𝐷 FuncCat 𝐸 ) ) |
107 |
5 9 106 101 7 11
|
funcf2 |
⊢ ( 𝜑 → ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) : ( 𝑋 𝐻 𝑍 ) ⟶ ( ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) ( 𝐷 Nat 𝐸 ) ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) ) ) |
108 |
107 13
|
ffvelrnd |
⊢ ( 𝜑 → ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ‘ 𝑅 ) ∈ ( ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) ( 𝐷 Nat 𝐸 ) ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) ) ) |
109 |
33 2 3 6 10 96 97 103 104 8 12 105 108 14
|
evlf2val |
⊢ ( 𝜑 → ( ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ‘ 𝑅 ) ( 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) , 𝑌 〉 ( 2nd ‘ ( 𝐷 evalF 𝐸 ) ) 〈 ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) , 𝑊 〉 ) 𝑆 ) = ( ( ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ‘ 𝑅 ) ‘ 𝑊 ) ( 〈 ( ( 1st ‘ ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) ) ‘ 𝑌 ) , ( ( 1st ‘ ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) ) ‘ 𝑊 ) 〉 ( comp ‘ 𝐸 ) ( ( 1st ‘ ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) ) ‘ 𝑊 ) ) ( ( 𝑌 ( 2nd ‘ ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) ) 𝑊 ) ‘ 𝑆 ) ) ) |
110 |
44 95 109
|
3eqtrd |
⊢ ( 𝜑 → ( 𝑅 ( 〈 𝑋 , 𝑌 〉 ( 2nd ‘ 𝐹 ) 〈 𝑍 , 𝑊 〉 ) 𝑆 ) = ( ( ( ( 𝑋 ( 2nd ‘ 𝐺 ) 𝑍 ) ‘ 𝑅 ) ‘ 𝑊 ) ( 〈 ( ( 1st ‘ ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) ) ‘ 𝑌 ) , ( ( 1st ‘ ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) ) ‘ 𝑊 ) 〉 ( comp ‘ 𝐸 ) ( ( 1st ‘ ( ( 1st ‘ 𝐺 ) ‘ 𝑍 ) ) ‘ 𝑊 ) ) ( ( 𝑌 ( 2nd ‘ ( ( 1st ‘ 𝐺 ) ‘ 𝑋 ) ) 𝑊 ) ‘ 𝑆 ) ) ) |