Theorem mrcssidd

Description: A set is contained in its Moore closure. Deduction form of mrcssid . (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	mrcssidd.1	⊢ ( 𝜑 → 𝐴 ∈ ( Moore ‘ 𝑋 ) )
	mrcssidd.2	⊢ 𝑁 = ( mrCls ‘ 𝐴 )
	mrcssidd.3	⊢ ( 𝜑 → 𝑈 ⊆ 𝑋 )
Assertion	mrcssidd	⊢ ( 𝜑 → 𝑈 ⊆ ( 𝑁 ‘ 𝑈 ) )

Proof

Step	Hyp	Ref	Expression
1		mrcssidd.1	⊢ ( 𝜑 → 𝐴 ∈ ( Moore ‘ 𝑋 ) )
2		mrcssidd.2	⊢ 𝑁 = ( mrCls ‘ 𝐴 )
3		mrcssidd.3	⊢ ( 𝜑 → 𝑈 ⊆ 𝑋 )
4	2	mrcssid	⊢ ( ( 𝐴 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝑋 ) → 𝑈 ⊆ ( 𝑁 ‘ 𝑈 ) )
5	1 3 4	syl2anc	⊢ ( 𝜑 → 𝑈 ⊆ ( 𝑁 ‘ 𝑈 ) )