Theorem igenss

Description: A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010)

	Ref	Expression
Hypotheses	igenval.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
	igenval.2	⊢ 𝑋 = ran 𝐺
Assertion	igenss	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ) → 𝑆 ⊆ ( 𝑅 IdlGen 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		igenval.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
2		igenval.2	⊢ 𝑋 = ran 𝐺
3		ssintub	⊢ 𝑆 ⊆ ∩ { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 }
4	1 2	igenval	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑅 IdlGen 𝑆 ) = ∩ { 𝑗 ∈ ( Idl ‘ 𝑅 ) ∣ 𝑆 ⊆ 𝑗 } )
5	3 4	sseqtrrid	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋 ) → 𝑆 ⊆ ( 𝑅 IdlGen 𝑆 ) )