Theorem mrcidmd

Description: Moore closure is idempotent. Deduction form of mrcidm . (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	mrcidmd	⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑁 ‘ 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	mrcidm	⊢ ( ( 𝐴 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝑋 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑈 ) ) = ( 𝑁 ‘ 𝑈 ) )
5	1 3 4	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑁 ‘ 𝑈 ) ) = ( 𝑁 ‘ 𝑈 ) )