Step |
Hyp |
Ref |
Expression |
1 |
|
rescabs2.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑉 ) |
2 |
|
rescabs2.j |
⊢ ( 𝜑 → 𝐽 Fn ( 𝑇 × 𝑇 ) ) |
3 |
|
rescabs2.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑊 ) |
4 |
|
rescabs2.t |
⊢ ( 𝜑 → 𝑇 ⊆ 𝑆 ) |
5 |
|
ressabs |
⊢ ( ( 𝑆 ∈ 𝑊 ∧ 𝑇 ⊆ 𝑆 ) → ( ( 𝐶 ↾s 𝑆 ) ↾s 𝑇 ) = ( 𝐶 ↾s 𝑇 ) ) |
6 |
3 4 5
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐶 ↾s 𝑆 ) ↾s 𝑇 ) = ( 𝐶 ↾s 𝑇 ) ) |
7 |
6
|
oveq1d |
⊢ ( 𝜑 → ( ( ( 𝐶 ↾s 𝑆 ) ↾s 𝑇 ) sSet 〈 ( Hom ‘ ndx ) , 𝐽 〉 ) = ( ( 𝐶 ↾s 𝑇 ) sSet 〈 ( Hom ‘ ndx ) , 𝐽 〉 ) ) |
8 |
|
eqid |
⊢ ( ( 𝐶 ↾s 𝑆 ) ↾cat 𝐽 ) = ( ( 𝐶 ↾s 𝑆 ) ↾cat 𝐽 ) |
9 |
|
ovexd |
⊢ ( 𝜑 → ( 𝐶 ↾s 𝑆 ) ∈ V ) |
10 |
3 4
|
ssexd |
⊢ ( 𝜑 → 𝑇 ∈ V ) |
11 |
8 9 10 2
|
rescval2 |
⊢ ( 𝜑 → ( ( 𝐶 ↾s 𝑆 ) ↾cat 𝐽 ) = ( ( ( 𝐶 ↾s 𝑆 ) ↾s 𝑇 ) sSet 〈 ( Hom ‘ ndx ) , 𝐽 〉 ) ) |
12 |
|
eqid |
⊢ ( 𝐶 ↾cat 𝐽 ) = ( 𝐶 ↾cat 𝐽 ) |
13 |
12 1 10 2
|
rescval2 |
⊢ ( 𝜑 → ( 𝐶 ↾cat 𝐽 ) = ( ( 𝐶 ↾s 𝑇 ) sSet 〈 ( Hom ‘ ndx ) , 𝐽 〉 ) ) |
14 |
7 11 13
|
3eqtr4d |
⊢ ( 𝜑 → ( ( 𝐶 ↾s 𝑆 ) ↾cat 𝐽 ) = ( 𝐶 ↾cat 𝐽 ) ) |