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Metamath Proof Explorer

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 )