Theorem mrissd

Description: An independent set of a Moore system is a subset of the base set. Deduction form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	mriss.1	⊢ 𝐼 = ( mrInd ‘ 𝐴 )
	mrissd.2	⊢ ( 𝜑 → 𝐴 ∈ ( Moore ‘ 𝑋 ) )
	mrissd.3	⊢ ( 𝜑 → 𝑆 ∈ 𝐼 )
Assertion	mrissd	⊢ ( 𝜑 → 𝑆 ⊆ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		mriss.1	⊢ 𝐼 = ( mrInd ‘ 𝐴 )
2		mrissd.2	⊢ ( 𝜑 → 𝐴 ∈ ( Moore ‘ 𝑋 ) )
3		mrissd.3	⊢ ( 𝜑 → 𝑆 ∈ 𝐼 )
4	1	mriss	⊢ ( ( 𝐴 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑆 ∈ 𝐼 ) → 𝑆 ⊆ 𝑋 )
5	2 3 4	syl2anc	⊢ ( 𝜑 → 𝑆 ⊆ 𝑋 )