Step |
Hyp |
Ref |
Expression |
1 |
|
resssetc.c |
⊢ 𝐶 = ( SetCat ‘ 𝑈 ) |
2 |
|
resssetc.d |
⊢ 𝐷 = ( SetCat ‘ 𝑉 ) |
3 |
|
resssetc.1 |
⊢ ( 𝜑 → 𝑈 ∈ 𝑊 ) |
4 |
|
resssetc.2 |
⊢ ( 𝜑 → 𝑉 ⊆ 𝑈 ) |
5 |
3 4
|
ssexd |
⊢ ( 𝜑 → 𝑉 ∈ V ) |
6 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → 𝑉 ∈ V ) |
7 |
|
eqid |
⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 ) |
8 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → 𝑥 ∈ 𝑉 ) |
9 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → 𝑦 ∈ 𝑉 ) |
10 |
2 6 7 8 9
|
setchom |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) = ( 𝑦 ↑m 𝑥 ) ) |
11 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → 𝑈 ∈ 𝑊 ) |
12 |
|
eqid |
⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 ) |
13 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → 𝑉 ⊆ 𝑈 ) |
14 |
13 8
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → 𝑥 ∈ 𝑈 ) |
15 |
13 9
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → 𝑦 ∈ 𝑈 ) |
16 |
1 11 12 14 15
|
setchom |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑦 ↑m 𝑥 ) ) |
17 |
|
eqid |
⊢ ( 𝐶 ↾s 𝑉 ) = ( 𝐶 ↾s 𝑉 ) |
18 |
17 12
|
resshom |
⊢ ( 𝑉 ∈ V → ( Hom ‘ 𝐶 ) = ( Hom ‘ ( 𝐶 ↾s 𝑉 ) ) ) |
19 |
5 18
|
syl |
⊢ ( 𝜑 → ( Hom ‘ 𝐶 ) = ( Hom ‘ ( 𝐶 ↾s 𝑉 ) ) ) |
20 |
19
|
oveqdr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ ( 𝐶 ↾s 𝑉 ) ) 𝑦 ) ) |
21 |
10 16 20
|
3eqtr2rd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑥 ( Hom ‘ ( 𝐶 ↾s 𝑉 ) ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) |
22 |
21
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 ( Hom ‘ ( 𝐶 ↾s 𝑉 ) ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) |
23 |
|
eqid |
⊢ ( Hom ‘ ( 𝐶 ↾s 𝑉 ) ) = ( Hom ‘ ( 𝐶 ↾s 𝑉 ) ) |
24 |
1 3
|
setcbas |
⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝐶 ) ) |
25 |
4 24
|
sseqtrd |
⊢ ( 𝜑 → 𝑉 ⊆ ( Base ‘ 𝐶 ) ) |
26 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
27 |
17 26
|
ressbas2 |
⊢ ( 𝑉 ⊆ ( Base ‘ 𝐶 ) → 𝑉 = ( Base ‘ ( 𝐶 ↾s 𝑉 ) ) ) |
28 |
25 27
|
syl |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ ( 𝐶 ↾s 𝑉 ) ) ) |
29 |
2 5
|
setcbas |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝐷 ) ) |
30 |
23 7 28 29
|
homfeq |
⊢ ( 𝜑 → ( ( Homf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( Homf ‘ 𝐷 ) ↔ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 ( Hom ‘ ( 𝐶 ↾s 𝑉 ) ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) ) |
31 |
22 30
|
mpbird |
⊢ ( 𝜑 → ( Homf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( Homf ‘ 𝐷 ) ) |
32 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑉 ∈ V ) |
33 |
|
eqid |
⊢ ( comp ‘ 𝐷 ) = ( comp ‘ 𝐷 ) |
34 |
|
simplr1 |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑥 ∈ 𝑉 ) |
35 |
|
simplr2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑦 ∈ 𝑉 ) |
36 |
|
simplr3 |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑧 ∈ 𝑉 ) |
37 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) |
38 |
2 32 7 34 35
|
elsetchom |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ↔ 𝑓 : 𝑥 ⟶ 𝑦 ) ) |
39 |
37 38
|
mpbid |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑓 : 𝑥 ⟶ 𝑦 ) |
40 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) |
41 |
2 32 7 35 36
|
elsetchom |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ↔ 𝑔 : 𝑦 ⟶ 𝑧 ) ) |
42 |
40 41
|
mpbid |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑔 : 𝑦 ⟶ 𝑧 ) |
43 |
2 32 33 34 35 36 39 42
|
setcco |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) = ( 𝑔 ∘ 𝑓 ) ) |
44 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑈 ∈ 𝑊 ) |
45 |
|
eqid |
⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) |
46 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑉 ⊆ 𝑈 ) |
47 |
46 34
|
sseldd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑥 ∈ 𝑈 ) |
48 |
46 35
|
sseldd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑦 ∈ 𝑈 ) |
49 |
46 36
|
sseldd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → 𝑧 ∈ 𝑈 ) |
50 |
1 44 45 47 48 49 39 42
|
setcco |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( 𝑔 ∘ 𝑓 ) ) |
51 |
17 45
|
ressco |
⊢ ( 𝑉 ∈ V → ( comp ‘ 𝐶 ) = ( comp ‘ ( 𝐶 ↾s 𝑉 ) ) ) |
52 |
5 51
|
syl |
⊢ ( 𝜑 → ( comp ‘ 𝐶 ) = ( comp ‘ ( 𝐶 ↾s 𝑉 ) ) ) |
53 |
52
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → ( comp ‘ 𝐶 ) = ( comp ‘ ( 𝐶 ↾s 𝑉 ) ) ) |
54 |
53
|
oveqd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) = ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( 𝐶 ↾s 𝑉 ) ) 𝑧 ) ) |
55 |
54
|
oveqd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( 𝐶 ↾s 𝑉 ) ) 𝑧 ) 𝑓 ) ) |
56 |
43 50 55
|
3eqtr2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ) ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( 𝐶 ↾s 𝑉 ) ) 𝑧 ) 𝑓 ) ) |
57 |
56
|
ralrimivva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( 𝐶 ↾s 𝑉 ) ) 𝑧 ) 𝑓 ) ) |
58 |
57
|
ralrimivvva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ∀ 𝑧 ∈ 𝑉 ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( 𝐶 ↾s 𝑉 ) ) 𝑧 ) 𝑓 ) ) |
59 |
|
eqid |
⊢ ( comp ‘ ( 𝐶 ↾s 𝑉 ) ) = ( comp ‘ ( 𝐶 ↾s 𝑉 ) ) |
60 |
31
|
eqcomd |
⊢ ( 𝜑 → ( Homf ‘ 𝐷 ) = ( Homf ‘ ( 𝐶 ↾s 𝑉 ) ) ) |
61 |
33 59 7 29 28 60
|
comfeq |
⊢ ( 𝜑 → ( ( compf ‘ 𝐷 ) = ( compf ‘ ( 𝐶 ↾s 𝑉 ) ) ↔ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ∀ 𝑧 ∈ 𝑉 ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ ( 𝐶 ↾s 𝑉 ) ) 𝑧 ) 𝑓 ) ) ) |
62 |
58 61
|
mpbird |
⊢ ( 𝜑 → ( compf ‘ 𝐷 ) = ( compf ‘ ( 𝐶 ↾s 𝑉 ) ) ) |
63 |
62
|
eqcomd |
⊢ ( 𝜑 → ( compf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( compf ‘ 𝐷 ) ) |
64 |
31 63
|
jca |
⊢ ( 𝜑 → ( ( Homf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( Homf ‘ 𝐷 ) ∧ ( compf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( compf ‘ 𝐷 ) ) ) |