Step |
Hyp |
Ref |
Expression |
1 |
|
funcres2b.a |
⊢ 𝐴 = ( Base ‘ 𝐶 ) |
2 |
|
funcres2b.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
3 |
|
funcres2b.r |
⊢ ( 𝜑 → 𝑅 ∈ ( Subcat ‘ 𝐷 ) ) |
4 |
|
funcres2b.s |
⊢ ( 𝜑 → 𝑅 Fn ( 𝑆 × 𝑆 ) ) |
5 |
|
funcres2b.1 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 ) |
6 |
|
funcres2b.2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 𝐺 𝑦 ) : 𝑌 ⟶ ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ) ) |
7 |
|
df-br |
⊢ ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ↔ 〈 𝐹 , 𝐺 〉 ∈ ( 𝐶 Func 𝐷 ) ) |
8 |
|
funcrcl |
⊢ ( 〈 𝐹 , 𝐺 〉 ∈ ( 𝐶 Func 𝐷 ) → ( 𝐶 ∈ Cat ∧ 𝐷 ∈ Cat ) ) |
9 |
7 8
|
sylbi |
⊢ ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 → ( 𝐶 ∈ Cat ∧ 𝐷 ∈ Cat ) ) |
10 |
9
|
simpld |
⊢ ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 → 𝐶 ∈ Cat ) |
11 |
10
|
a1i |
⊢ ( 𝜑 → ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 → 𝐶 ∈ Cat ) ) |
12 |
|
df-br |
⊢ ( 𝐹 ( 𝐶 Func ( 𝐷 ↾cat 𝑅 ) ) 𝐺 ↔ 〈 𝐹 , 𝐺 〉 ∈ ( 𝐶 Func ( 𝐷 ↾cat 𝑅 ) ) ) |
13 |
|
funcrcl |
⊢ ( 〈 𝐹 , 𝐺 〉 ∈ ( 𝐶 Func ( 𝐷 ↾cat 𝑅 ) ) → ( 𝐶 ∈ Cat ∧ ( 𝐷 ↾cat 𝑅 ) ∈ Cat ) ) |
14 |
12 13
|
sylbi |
⊢ ( 𝐹 ( 𝐶 Func ( 𝐷 ↾cat 𝑅 ) ) 𝐺 → ( 𝐶 ∈ Cat ∧ ( 𝐷 ↾cat 𝑅 ) ∈ Cat ) ) |
15 |
14
|
simpld |
⊢ ( 𝐹 ( 𝐶 Func ( 𝐷 ↾cat 𝑅 ) ) 𝐺 → 𝐶 ∈ Cat ) |
16 |
15
|
a1i |
⊢ ( 𝜑 → ( 𝐹 ( 𝐶 Func ( 𝐷 ↾cat 𝑅 ) ) 𝐺 → 𝐶 ∈ Cat ) ) |
17 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
18 |
3 4 17
|
subcss1 |
⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝐷 ) ) |
19 |
5 18
|
fssd |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ( Base ‘ 𝐷 ) ) |
20 |
|
eqid |
⊢ ( 𝐷 ↾cat 𝑅 ) = ( 𝐷 ↾cat 𝑅 ) |
21 |
|
subcrcl |
⊢ ( 𝑅 ∈ ( Subcat ‘ 𝐷 ) → 𝐷 ∈ Cat ) |
22 |
3 21
|
syl |
⊢ ( 𝜑 → 𝐷 ∈ Cat ) |
23 |
20 17 22 4 18
|
rescbas |
⊢ ( 𝜑 → 𝑆 = ( Base ‘ ( 𝐷 ↾cat 𝑅 ) ) ) |
24 |
23
|
feq3d |
⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ 𝑆 ↔ 𝐹 : 𝐴 ⟶ ( Base ‘ ( 𝐷 ↾cat 𝑅 ) ) ) ) |
25 |
5 24
|
mpbid |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ( Base ‘ ( 𝐷 ↾cat 𝑅 ) ) ) |
26 |
19 25
|
2thd |
⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ ( Base ‘ 𝐷 ) ↔ 𝐹 : 𝐴 ⟶ ( Base ‘ ( 𝐷 ↾cat 𝑅 ) ) ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → ( 𝐹 : 𝐴 ⟶ ( Base ‘ 𝐷 ) ↔ 𝐹 : 𝐴 ⟶ ( Base ‘ ( 𝐷 ↾cat 𝑅 ) ) ) ) |
28 |
6
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 𝐺 𝑦 ) : 𝑌 ⟶ ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ) ) |
29 |
28
|
frnd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ran ( 𝑥 𝐺 𝑦 ) ⊆ ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ) ) |
30 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → 𝑅 ∈ ( Subcat ‘ 𝐷 ) ) |
31 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → 𝑅 Fn ( 𝑆 × 𝑆 ) ) |
32 |
|
eqid |
⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 ) |
33 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → 𝐹 : 𝐴 ⟶ 𝑆 ) |
34 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → 𝑥 ∈ 𝐴 ) |
35 |
33 34
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
36 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → 𝑦 ∈ 𝐴 ) |
37 |
33 36
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑆 ) |
38 |
30 31 32 35 37
|
subcss2 |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ) |
39 |
29 38
|
sstrd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ran ( 𝑥 𝐺 𝑦 ) ⊆ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ) |
40 |
39 29
|
2thd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ran ( 𝑥 𝐺 𝑦 ) ⊆ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ran ( 𝑥 𝐺 𝑦 ) ⊆ ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ) ) ) |
41 |
40
|
anbi2d |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( ( 𝑥 𝐺 𝑦 ) Fn ( 𝑥 𝐻 𝑦 ) ∧ ran ( 𝑥 𝐺 𝑦 ) ⊆ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ) ↔ ( ( 𝑥 𝐺 𝑦 ) Fn ( 𝑥 𝐻 𝑦 ) ∧ ran ( 𝑥 𝐺 𝑦 ) ⊆ ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
42 |
|
df-f |
⊢ ( ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( 𝑥 𝐺 𝑦 ) Fn ( 𝑥 𝐻 𝑦 ) ∧ ran ( 𝑥 𝐺 𝑦 ) ⊆ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
43 |
|
df-f |
⊢ ( ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( 𝑥 𝐺 𝑦 ) Fn ( 𝑥 𝐻 𝑦 ) ∧ ran ( 𝑥 𝐺 𝑦 ) ⊆ ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ) ) ) |
44 |
41 42 43
|
3bitr4g |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ) ) ) |
45 |
20 17 22 4 18
|
reschom |
⊢ ( 𝜑 → 𝑅 = ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ) |
46 |
45
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → 𝑅 = ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ) |
47 |
46
|
oveqd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ) |
48 |
47
|
feq3d |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
49 |
44 48
|
bitrd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
50 |
49
|
ralrimivva |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
51 |
|
fveq2 |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 〈 𝑥 , 𝑦 〉 ) ) |
52 |
|
df-ov |
⊢ ( 𝑥 𝐺 𝑦 ) = ( 𝐺 ‘ 〈 𝑥 , 𝑦 〉 ) |
53 |
51 52
|
eqtr4di |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 𝐺 ‘ 𝑧 ) = ( 𝑥 𝐺 𝑦 ) ) |
54 |
|
vex |
⊢ 𝑥 ∈ V |
55 |
|
vex |
⊢ 𝑦 ∈ V |
56 |
54 55
|
op1std |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 1st ‘ 𝑧 ) = 𝑥 ) |
57 |
56
|
fveq2d |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
58 |
54 55
|
op2ndd |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 2nd ‘ 𝑧 ) = 𝑦 ) |
59 |
58
|
fveq2d |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑦 ) ) |
60 |
57 59
|
oveq12d |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) = ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ) |
61 |
|
fveq2 |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 𝐻 ‘ 𝑧 ) = ( 𝐻 ‘ 〈 𝑥 , 𝑦 〉 ) ) |
62 |
|
df-ov |
⊢ ( 𝑥 𝐻 𝑦 ) = ( 𝐻 ‘ 〈 𝑥 , 𝑦 〉 ) |
63 |
61 62
|
eqtr4di |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 𝐻 ‘ 𝑧 ) = ( 𝑥 𝐻 𝑦 ) ) |
64 |
60 63
|
oveq12d |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) = ( ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ↑m ( 𝑥 𝐻 𝑦 ) ) ) |
65 |
53 64
|
eleq12d |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ ( 𝑥 𝐺 𝑦 ) ∈ ( ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ↑m ( 𝑥 𝐻 𝑦 ) ) ) ) |
66 |
|
ovex |
⊢ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ∈ V |
67 |
|
ovex |
⊢ ( 𝑥 𝐻 𝑦 ) ∈ V |
68 |
66 67
|
elmap |
⊢ ( ( 𝑥 𝐺 𝑦 ) ∈ ( ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ↑m ( 𝑥 𝐻 𝑦 ) ) ↔ ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ) |
69 |
65 68
|
bitrdi |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
70 |
57 59
|
oveq12d |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) = ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ) |
71 |
70 63
|
oveq12d |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) = ( ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ↑m ( 𝑥 𝐻 𝑦 ) ) ) |
72 |
53 71
|
eleq12d |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ ( 𝑥 𝐺 𝑦 ) ∈ ( ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ↑m ( 𝑥 𝐻 𝑦 ) ) ) ) |
73 |
|
ovex |
⊢ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ∈ V |
74 |
73 67
|
elmap |
⊢ ( ( 𝑥 𝐺 𝑦 ) ∈ ( ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ↑m ( 𝑥 𝐻 𝑦 ) ) ↔ ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ) |
75 |
72 74
|
bitrdi |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
76 |
69 75
|
bibi12d |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( ( ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ) ↔ ( ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
77 |
76
|
ralxp |
⊢ ( ∀ 𝑧 ∈ ( 𝐴 × 𝐴 ) ( ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑥 𝐺 𝑦 ) : ( 𝑥 𝐻 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑥 ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
78 |
50 77
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → ∀ 𝑧 ∈ ( 𝐴 × 𝐴 ) ( ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ) ) |
79 |
|
ralbi |
⊢ ( ∀ 𝑧 ∈ ( 𝐴 × 𝐴 ) ( ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ) → ( ∀ 𝑧 ∈ ( 𝐴 × 𝐴 ) ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ ∀ 𝑧 ∈ ( 𝐴 × 𝐴 ) ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ) ) |
80 |
78 79
|
syl |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → ( ∀ 𝑧 ∈ ( 𝐴 × 𝐴 ) ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ ∀ 𝑧 ∈ ( 𝐴 × 𝐴 ) ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ) ) |
81 |
80
|
3anbi3d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → ( ( 𝐺 ∈ V ∧ 𝐺 Fn ( 𝐴 × 𝐴 ) ∧ ∀ 𝑧 ∈ ( 𝐴 × 𝐴 ) ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ) ↔ ( 𝐺 ∈ V ∧ 𝐺 Fn ( 𝐴 × 𝐴 ) ∧ ∀ 𝑧 ∈ ( 𝐴 × 𝐴 ) ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ) ) ) |
82 |
|
elixp2 |
⊢ ( 𝐺 ∈ X 𝑧 ∈ ( 𝐴 × 𝐴 ) ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ ( 𝐺 ∈ V ∧ 𝐺 Fn ( 𝐴 × 𝐴 ) ∧ ∀ 𝑧 ∈ ( 𝐴 × 𝐴 ) ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ) ) |
83 |
|
elixp2 |
⊢ ( 𝐺 ∈ X 𝑧 ∈ ( 𝐴 × 𝐴 ) ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ ( 𝐺 ∈ V ∧ 𝐺 Fn ( 𝐴 × 𝐴 ) ∧ ∀ 𝑧 ∈ ( 𝐴 × 𝐴 ) ( 𝐺 ‘ 𝑧 ) ∈ ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ) ) |
84 |
81 82 83
|
3bitr4g |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → ( 𝐺 ∈ X 𝑧 ∈ ( 𝐴 × 𝐴 ) ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ↔ 𝐺 ∈ X 𝑧 ∈ ( 𝐴 × 𝐴 ) ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ) ) |
85 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ 𝑥 ∈ 𝐴 ) → 𝑅 ∈ ( Subcat ‘ 𝐷 ) ) |
86 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ 𝑥 ∈ 𝐴 ) → 𝑅 Fn ( 𝑆 × 𝑆 ) ) |
87 |
|
eqid |
⊢ ( Id ‘ 𝐷 ) = ( Id ‘ 𝐷 ) |
88 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → 𝐹 : 𝐴 ⟶ 𝑆 ) |
89 |
88
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝑆 ) |
90 |
20 85 86 87 89
|
subcid |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ 𝑥 ∈ 𝐴 ) → ( ( Id ‘ 𝐷 ) ‘ ( 𝐹 ‘ 𝑥 ) ) = ( ( Id ‘ ( 𝐷 ↾cat 𝑅 ) ) ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
91 |
90
|
eqeq2d |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ 𝑥 ∈ 𝐴 ) → ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝐷 ) ‘ ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( 𝑥 𝐺 𝑥 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ) = ( ( Id ‘ ( 𝐷 ↾cat 𝑅 ) ) ‘ ( 𝐹 ‘ 𝑥 ) ) ) ) |
92 |
|
eqid |
⊢ ( comp ‘ 𝐷 ) = ( comp ‘ 𝐷 ) |
93 |
20 17 22 4 18 92
|
rescco |
⊢ ( 𝜑 → ( comp ‘ 𝐷 ) = ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) ) |
94 |
93
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ 𝑥 ∈ 𝐴 ) → ( comp ‘ 𝐷 ) = ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) ) |
95 |
94
|
oveqd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ 𝑥 ∈ 𝐴 ) → ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐷 ) ( 𝐹 ‘ 𝑧 ) ) = ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑧 ) ) ) |
96 |
95
|
oveqd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ 𝑥 ∈ 𝐴 ) → ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐷 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ) |
97 |
96
|
eqeq2d |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ 𝑥 ∈ 𝐴 ) → ( ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐷 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ↔ ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ) ) |
98 |
97
|
2ralbidv |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐷 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ) ) |
99 |
98
|
2ralbidv |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ 𝑥 ∈ 𝐴 ) → ( ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐷 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ↔ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ) ) |
100 |
91 99
|
anbi12d |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ Cat ) ∧ 𝑥 ∈ 𝐴 ) → ( ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝐷 ) ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐷 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ) ↔ ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ) = ( ( Id ‘ ( 𝐷 ↾cat 𝑅 ) ) ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ) ) ) |
101 |
100
|
ralbidva |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → ( ∀ 𝑥 ∈ 𝐴 ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝐷 ) ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐷 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ) ↔ ∀ 𝑥 ∈ 𝐴 ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ) = ( ( Id ‘ ( 𝐷 ↾cat 𝑅 ) ) ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ) ) ) |
102 |
27 84 101
|
3anbi123d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → ( ( 𝐹 : 𝐴 ⟶ ( Base ‘ 𝐷 ) ∧ 𝐺 ∈ X 𝑧 ∈ ( 𝐴 × 𝐴 ) ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝐷 ) ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐷 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ) ) ↔ ( 𝐹 : 𝐴 ⟶ ( Base ‘ ( 𝐷 ↾cat 𝑅 ) ) ∧ 𝐺 ∈ X 𝑧 ∈ ( 𝐴 × 𝐴 ) ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ) = ( ( Id ‘ ( 𝐷 ↾cat 𝑅 ) ) ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ) ) ) ) |
103 |
|
eqid |
⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 ) |
104 |
|
eqid |
⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) |
105 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → 𝐶 ∈ Cat ) |
106 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → 𝐷 ∈ Cat ) |
107 |
1 17 2 32 103 87 104 92 105 106
|
isfunc |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ↔ ( 𝐹 : 𝐴 ⟶ ( Base ‘ 𝐷 ) ∧ 𝐺 ∈ X 𝑧 ∈ ( 𝐴 × 𝐴 ) ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐷 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝐷 ) ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐷 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ) ) ) ) |
108 |
|
eqid |
⊢ ( Base ‘ ( 𝐷 ↾cat 𝑅 ) ) = ( Base ‘ ( 𝐷 ↾cat 𝑅 ) ) |
109 |
|
eqid |
⊢ ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) = ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) |
110 |
|
eqid |
⊢ ( Id ‘ ( 𝐷 ↾cat 𝑅 ) ) = ( Id ‘ ( 𝐷 ↾cat 𝑅 ) ) |
111 |
|
eqid |
⊢ ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) = ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) |
112 |
20 3
|
subccat |
⊢ ( 𝜑 → ( 𝐷 ↾cat 𝑅 ) ∈ Cat ) |
113 |
112
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → ( 𝐷 ↾cat 𝑅 ) ∈ Cat ) |
114 |
1 108 2 109 103 110 104 111 105 113
|
isfunc |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → ( 𝐹 ( 𝐶 Func ( 𝐷 ↾cat 𝑅 ) ) 𝐺 ↔ ( 𝐹 : 𝐴 ⟶ ( Base ‘ ( 𝐷 ↾cat 𝑅 ) ) ∧ 𝐺 ∈ X 𝑧 ∈ ( 𝐴 × 𝐴 ) ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( 𝐻 ‘ 𝑧 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( ( Id ‘ 𝐶 ) ‘ 𝑥 ) ) = ( ( Id ‘ ( 𝐷 ↾cat 𝑅 ) ) ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ∀ 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐻 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑔 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ ( 𝐷 ↾cat 𝑅 ) ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑓 ) ) ) ) ) ) |
115 |
102 107 114
|
3bitr4d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ Cat ) → ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ↔ 𝐹 ( 𝐶 Func ( 𝐷 ↾cat 𝑅 ) ) 𝐺 ) ) |
116 |
115
|
ex |
⊢ ( 𝜑 → ( 𝐶 ∈ Cat → ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ↔ 𝐹 ( 𝐶 Func ( 𝐷 ↾cat 𝑅 ) ) 𝐺 ) ) ) |
117 |
11 16 116
|
pm5.21ndd |
⊢ ( 𝜑 → ( 𝐹 ( 𝐶 Func 𝐷 ) 𝐺 ↔ 𝐹 ( 𝐶 Func ( 𝐷 ↾cat 𝑅 ) ) 𝐺 ) ) |