Metamath Proof Explorer
Description: Moore closure preserves subset ordering. Deduction form of mrcss .
(Contributed by David Moews, 1-May-2017)
|
|
Ref |
Expression |
|
Hypotheses |
mrcssd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ( Moore ‘ 𝑋 ) ) |
|
|
mrcssd.2 |
⊢ 𝑁 = ( mrCls ‘ 𝐴 ) |
|
|
mrcssd.3 |
⊢ ( 𝜑 → 𝑈 ⊆ 𝑉 ) |
|
|
mrcssd.4 |
⊢ ( 𝜑 → 𝑉 ⊆ 𝑋 ) |
|
Assertion |
mrcssd |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝑈 ) ⊆ ( 𝑁 ‘ 𝑉 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
mrcssd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ( Moore ‘ 𝑋 ) ) |
2 |
|
mrcssd.2 |
⊢ 𝑁 = ( mrCls ‘ 𝐴 ) |
3 |
|
mrcssd.3 |
⊢ ( 𝜑 → 𝑈 ⊆ 𝑉 ) |
4 |
|
mrcssd.4 |
⊢ ( 𝜑 → 𝑉 ⊆ 𝑋 ) |
5 |
2
|
mrcss |
⊢ ( ( 𝐴 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝑉 ∧ 𝑉 ⊆ 𝑋 ) → ( 𝑁 ‘ 𝑈 ) ⊆ ( 𝑁 ‘ 𝑉 ) ) |
6 |
1 3 4 5
|
syl3anc |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝑈 ) ⊆ ( 𝑁 ‘ 𝑉 ) ) |