Theorem subsubrng2

Description: The set of subrings of a subring are the smaller subrings. (Contributed by AV, 15-Feb-2025)

	Ref	Expression
Hypothesis	subsubrng.s	⊢ 𝑆 = ( 𝑅 ↾s 𝐴 )
Assertion	subsubrng2	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → ( SubRng ‘ 𝑆 ) = ( ( SubRng ‘ 𝑅 ) ∩ 𝒫 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		subsubrng.s	⊢ 𝑆 = ( 𝑅 ↾s 𝐴 )
2	1	subsubrng	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → ( 𝑎 ∈ ( SubRng ‘ 𝑆 ) ↔ ( 𝑎 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑎 ⊆ 𝐴 ) ) )
3		elin	⊢ ( 𝑎 ∈ ( ( SubRng ‘ 𝑅 ) ∩ 𝒫 𝐴 ) ↔ ( 𝑎 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑎 ∈ 𝒫 𝐴 ) )
4		velpw	⊢ ( 𝑎 ∈ 𝒫 𝐴 ↔ 𝑎 ⊆ 𝐴 )
5	4	anbi2i	⊢ ( ( 𝑎 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑎 ∈ 𝒫 𝐴 ) ↔ ( 𝑎 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑎 ⊆ 𝐴 ) )
6	3 5	bitr2i	⊢ ( ( 𝑎 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑎 ⊆ 𝐴 ) ↔ 𝑎 ∈ ( ( SubRng ‘ 𝑅 ) ∩ 𝒫 𝐴 ) )
7	2 6	bitrdi	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → ( 𝑎 ∈ ( SubRng ‘ 𝑆 ) ↔ 𝑎 ∈ ( ( SubRng ‘ 𝑅 ) ∩ 𝒫 𝐴 ) ) )
8	7	eqrdv	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → ( SubRng ‘ 𝑆 ) = ( ( SubRng ‘ 𝑅 ) ∩ 𝒫 𝐴 ) )