Theorem mrcssvd

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 )