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 ) )