Theorem ismri2d

Description: Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	ismri2.1	⊢ 𝑁 = ( mrCls ‘ 𝐴 )
	ismri2.2	⊢ 𝐼 = ( mrInd ‘ 𝐴 )
	ismri2d.3	⊢ ( 𝜑 → 𝐴 ∈ ( Moore ‘ 𝑋 ) )
	ismri2d.4	⊢ ( 𝜑 → 𝑆 ⊆ 𝑋 )
Assertion	ismri2d	⊢ ( 𝜑 → ( 𝑆 ∈ 𝐼 ↔ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ismri2.1	⊢ 𝑁 = ( mrCls ‘ 𝐴 )
2		ismri2.2	⊢ 𝐼 = ( mrInd ‘ 𝐴 )
3		ismri2d.3	⊢ ( 𝜑 → 𝐴 ∈ ( Moore ‘ 𝑋 ) )
4		ismri2d.4	⊢ ( 𝜑 → 𝑆 ⊆ 𝑋 )
5	1 2	ismri2	⊢ ( ( 𝐴 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑆 ∈ 𝐼 ↔ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) )
6	3 4 5	syl2anc	⊢ ( 𝜑 → ( 𝑆 ∈ 𝐼 ↔ ∀ 𝑥 ∈ 𝑆 ¬ 𝑥 ∈ ( 𝑁 ‘ ( 𝑆 ∖ { 𝑥 } ) ) ) )