Step |
Hyp |
Ref |
Expression |
1 |
|
rescbas.d |
⊢ 𝐷 = ( 𝐶 ↾cat 𝐻 ) |
2 |
|
rescbas.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
3 |
|
rescbas.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑉 ) |
4 |
|
rescbas.h |
⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) ) |
5 |
|
rescbas.s |
⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 ) |
6 |
1 2 3 4 5
|
reschom |
⊢ ( 𝜑 → 𝐻 = ( Hom ‘ 𝐷 ) ) |
7 |
1 2 3 4 5
|
rescbas |
⊢ ( 𝜑 → 𝑆 = ( Base ‘ 𝐷 ) ) |
8 |
7
|
sqxpeqd |
⊢ ( 𝜑 → ( 𝑆 × 𝑆 ) = ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐷 ) ) ) |
9 |
6 8
|
fneq12d |
⊢ ( 𝜑 → ( 𝐻 Fn ( 𝑆 × 𝑆 ) ↔ ( Hom ‘ 𝐷 ) Fn ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐷 ) ) ) ) |
10 |
4 9
|
mpbid |
⊢ ( 𝜑 → ( Hom ‘ 𝐷 ) Fn ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐷 ) ) ) |
11 |
|
fnov |
⊢ ( ( Hom ‘ 𝐷 ) Fn ( ( Base ‘ 𝐷 ) × ( Base ‘ 𝐷 ) ) ↔ ( Hom ‘ 𝐷 ) = ( 𝑥 ∈ ( Base ‘ 𝐷 ) , 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) ) |
12 |
10 11
|
sylib |
⊢ ( 𝜑 → ( Hom ‘ 𝐷 ) = ( 𝑥 ∈ ( Base ‘ 𝐷 ) , 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) ) |
13 |
6 12
|
eqtrd |
⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ ( Base ‘ 𝐷 ) , 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) ) |
14 |
|
eqid |
⊢ ( Homf ‘ 𝐷 ) = ( Homf ‘ 𝐷 ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 ) |
16 |
|
eqid |
⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 ) |
17 |
14 15 16
|
homffval |
⊢ ( Homf ‘ 𝐷 ) = ( 𝑥 ∈ ( Base ‘ 𝐷 ) , 𝑦 ∈ ( Base ‘ 𝐷 ) ↦ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) |
18 |
13 17
|
eqtr4di |
⊢ ( 𝜑 → 𝐻 = ( Homf ‘ 𝐷 ) ) |