Metamath Proof Explorer
Description: The Moore closure of a set is a subset of the base. Deduction form of
mrcssv . (Contributed by David Moews, 1-May-2017)
|
|
Ref |
Expression |
|
Hypotheses |
mrcssd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ( Moore ‘ 𝑋 ) ) |
|
|
mrcssd.2 |
⊢ 𝑁 = ( mrCls ‘ 𝐴 ) |
|
Assertion |
mrcssvd |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝐵 ) ⊆ 𝑋 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
mrcssd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ( Moore ‘ 𝑋 ) ) |
2 |
|
mrcssd.2 |
⊢ 𝑁 = ( mrCls ‘ 𝐴 ) |
3 |
2
|
mrcssv |
⊢ ( 𝐴 ∈ ( Moore ‘ 𝑋 ) → ( 𝑁 ‘ 𝐵 ) ⊆ 𝑋 ) |
4 |
1 3
|
syl |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝐵 ) ⊆ 𝑋 ) |