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Metamath Proof Explorer

Theorem subsubrg2

Description: The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015)

Ref Expression
Hypothesis subsubrg.s ⊢ 𝑆 = ( 𝑅 ↾s 𝐴 )
Assertion subsubrg2 ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) → ( SubRing ‘ 𝑆 ) = ( ( SubRing ‘ 𝑅 ) ∩ 𝒫 𝐴 ) )

Proof

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