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Metamath Proof Explorer

Theorem mrcssd

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 ) )