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 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( Homf ‘ 𝐸 ) = ( Homf ‘ 𝐸 ) ) |
6 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( compf ‘ 𝐸 ) = ( compf ‘ 𝐸 ) ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
8 |
|
eqid |
⊢ ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐶 ) |
9 |
1
|
setccat |
⊢ ( 𝑈 ∈ 𝑊 → 𝐶 ∈ Cat ) |
10 |
3 9
|
syl |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
11 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → 𝐶 ∈ Cat ) |
12 |
1 3
|
setcbas |
⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝐶 ) ) |
13 |
4 12
|
sseqtrd |
⊢ ( 𝜑 → 𝑉 ⊆ ( Base ‘ 𝐶 ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → 𝑉 ⊆ ( Base ‘ 𝐶 ) ) |
15 |
|
eqid |
⊢ ( 𝐶 ↾s 𝑉 ) = ( 𝐶 ↾s 𝑉 ) |
16 |
|
eqid |
⊢ ( 𝐶 ↾cat ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ) = ( 𝐶 ↾cat ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ) |
17 |
7 8 11 14 15 16
|
fullresc |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( ( Homf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( Homf ‘ ( 𝐶 ↾cat ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ) ) ∧ ( compf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( compf ‘ ( 𝐶 ↾cat ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ) ) ) ) |
18 |
17
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( Homf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( Homf ‘ ( 𝐶 ↾cat ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ) ) ) |
19 |
1 2 3 4
|
resssetc |
⊢ ( 𝜑 → ( ( Homf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( Homf ‘ 𝐷 ) ∧ ( compf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( compf ‘ 𝐷 ) ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( ( Homf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( Homf ‘ 𝐷 ) ∧ ( compf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( compf ‘ 𝐷 ) ) ) |
21 |
20
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( Homf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( Homf ‘ 𝐷 ) ) |
22 |
18 21
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( Homf ‘ ( 𝐶 ↾cat ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ) ) = ( Homf ‘ 𝐷 ) ) |
23 |
17
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( compf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( compf ‘ ( 𝐶 ↾cat ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ) ) ) |
24 |
20
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( compf ‘ ( 𝐶 ↾s 𝑉 ) ) = ( compf ‘ 𝐷 ) ) |
25 |
23 24
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( compf ‘ ( 𝐶 ↾cat ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ) ) = ( compf ‘ 𝐷 ) ) |
26 |
|
funcrcl |
⊢ ( 𝑓 ∈ ( 𝐸 Func 𝐷 ) → ( 𝐸 ∈ Cat ∧ 𝐷 ∈ Cat ) ) |
27 |
26
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( 𝐸 ∈ Cat ∧ 𝐷 ∈ Cat ) ) |
28 |
27
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → 𝐸 ∈ Cat ) |
29 |
7 8 11 14
|
fullsubc |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ∈ ( Subcat ‘ 𝐶 ) ) |
30 |
16 29
|
subccat |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( 𝐶 ↾cat ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ) ∈ Cat ) |
31 |
27
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → 𝐷 ∈ Cat ) |
32 |
5 6 22 25 28 28 30 31
|
funcpropd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( 𝐸 Func ( 𝐶 ↾cat ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ) ) = ( 𝐸 Func 𝐷 ) ) |
33 |
|
funcres2 |
⊢ ( ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ∈ ( Subcat ‘ 𝐶 ) → ( 𝐸 Func ( 𝐶 ↾cat ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ) ) ⊆ ( 𝐸 Func 𝐶 ) ) |
34 |
29 33
|
syl |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( 𝐸 Func ( 𝐶 ↾cat ( ( Homf ‘ 𝐶 ) ↾ ( 𝑉 × 𝑉 ) ) ) ) ⊆ ( 𝐸 Func 𝐶 ) ) |
35 |
32 34
|
eqsstrrd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → ( 𝐸 Func 𝐷 ) ⊆ ( 𝐸 Func 𝐶 ) ) |
36 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) |
37 |
35 36
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝐸 Func 𝐷 ) ) → 𝑓 ∈ ( 𝐸 Func 𝐶 ) ) |
38 |
37
|
ex |
⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝐸 Func 𝐷 ) → 𝑓 ∈ ( 𝐸 Func 𝐶 ) ) ) |
39 |
38
|
ssrdv |
⊢ ( 𝜑 → ( 𝐸 Func 𝐷 ) ⊆ ( 𝐸 Func 𝐶 ) ) |