Step |
Hyp |
Ref |
Expression |
1 |
|
rescval.1 |
⊢ 𝐷 = ( 𝐶 ↾cat 𝐻 ) |
2 |
|
elex |
⊢ ( 𝐶 ∈ 𝑉 → 𝐶 ∈ V ) |
3 |
|
elex |
⊢ ( 𝐻 ∈ 𝑊 → 𝐻 ∈ V ) |
4 |
|
simpl |
⊢ ( ( 𝑐 = 𝐶 ∧ ℎ = 𝐻 ) → 𝑐 = 𝐶 ) |
5 |
|
simpr |
⊢ ( ( 𝑐 = 𝐶 ∧ ℎ = 𝐻 ) → ℎ = 𝐻 ) |
6 |
5
|
dmeqd |
⊢ ( ( 𝑐 = 𝐶 ∧ ℎ = 𝐻 ) → dom ℎ = dom 𝐻 ) |
7 |
6
|
dmeqd |
⊢ ( ( 𝑐 = 𝐶 ∧ ℎ = 𝐻 ) → dom dom ℎ = dom dom 𝐻 ) |
8 |
4 7
|
oveq12d |
⊢ ( ( 𝑐 = 𝐶 ∧ ℎ = 𝐻 ) → ( 𝑐 ↾s dom dom ℎ ) = ( 𝐶 ↾s dom dom 𝐻 ) ) |
9 |
5
|
opeq2d |
⊢ ( ( 𝑐 = 𝐶 ∧ ℎ = 𝐻 ) → 〈 ( Hom ‘ ndx ) , ℎ 〉 = 〈 ( Hom ‘ ndx ) , 𝐻 〉 ) |
10 |
8 9
|
oveq12d |
⊢ ( ( 𝑐 = 𝐶 ∧ ℎ = 𝐻 ) → ( ( 𝑐 ↾s dom dom ℎ ) sSet 〈 ( Hom ‘ ndx ) , ℎ 〉 ) = ( ( 𝐶 ↾s dom dom 𝐻 ) sSet 〈 ( Hom ‘ ndx ) , 𝐻 〉 ) ) |
11 |
|
df-resc |
⊢ ↾cat = ( 𝑐 ∈ V , ℎ ∈ V ↦ ( ( 𝑐 ↾s dom dom ℎ ) sSet 〈 ( Hom ‘ ndx ) , ℎ 〉 ) ) |
12 |
|
ovex |
⊢ ( ( 𝐶 ↾s dom dom 𝐻 ) sSet 〈 ( Hom ‘ ndx ) , 𝐻 〉 ) ∈ V |
13 |
10 11 12
|
ovmpoa |
⊢ ( ( 𝐶 ∈ V ∧ 𝐻 ∈ V ) → ( 𝐶 ↾cat 𝐻 ) = ( ( 𝐶 ↾s dom dom 𝐻 ) sSet 〈 ( Hom ‘ ndx ) , 𝐻 〉 ) ) |
14 |
2 3 13
|
syl2an |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊 ) → ( 𝐶 ↾cat 𝐻 ) = ( ( 𝐶 ↾s dom dom 𝐻 ) sSet 〈 ( Hom ‘ ndx ) , 𝐻 〉 ) ) |
15 |
1 14
|
eqtrid |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊 ) → 𝐷 = ( ( 𝐶 ↾s dom dom 𝐻 ) sSet 〈 ( Hom ‘ ndx ) , 𝐻 〉 ) ) |