Metamath Proof Explorer
Description: comp is unaffected by restriction. (Contributed by Mario
Carneiro, 5-Jan-2017)
|
|
Ref |
Expression |
|
Hypotheses |
resshom.1 |
⊢ 𝐷 = ( 𝐶 ↾s 𝐴 ) |
|
|
ressco.2 |
⊢ · = ( comp ‘ 𝐶 ) |
|
Assertion |
ressco |
⊢ ( 𝐴 ∈ 𝑉 → · = ( comp ‘ 𝐷 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
resshom.1 |
⊢ 𝐷 = ( 𝐶 ↾s 𝐴 ) |
2 |
|
ressco.2 |
⊢ · = ( comp ‘ 𝐶 ) |
3 |
|
df-cco |
⊢ comp = Slot ; 1 5 |
4 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
5 |
|
5nn |
⊢ 5 ∈ ℕ |
6 |
4 5
|
decnncl |
⊢ ; 1 5 ∈ ℕ |
7 |
|
1nn |
⊢ 1 ∈ ℕ |
8 |
|
5nn0 |
⊢ 5 ∈ ℕ0 |
9 |
|
1lt10 |
⊢ 1 < ; 1 0 |
10 |
7 8 4 9
|
declti |
⊢ 1 < ; 1 5 |
11 |
1 2 3 6 10
|
resslem |
⊢ ( 𝐴 ∈ 𝑉 → · = ( comp ‘ 𝐷 ) ) |