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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 e. ( SubRing ` R ) -> ( SubRing ` S ) = ( ( SubRing ` R ) i^i ~P A ) )

Proof

Step Hyp Ref Expression
1 subsubrg.s 
 |-  S = ( R |`s A )
2 1 subsubrg 
 |-  ( A e. ( SubRing ` R ) -> ( a e. ( SubRing ` S ) <-> ( a e. ( SubRing ` R ) /\ a C_ A ) ) )
3 elin 
 |-  ( a e. ( ( SubRing ` R ) i^i ~P A ) <-> ( a e. ( SubRing ` R ) /\ a e. ~P A ) )
4 velpw 
 |-  ( a e. ~P A <-> a C_ A )
5 4 anbi2i 
 |-  ( ( a e. ( SubRing ` R ) /\ a e. ~P A ) <-> ( a e. ( SubRing ` R ) /\ a C_ A ) )
6 3 5 bitr2i 
 |-  ( ( a e. ( SubRing ` R ) /\ a C_ A ) <-> a e. ( ( SubRing ` R ) i^i ~P A ) )
7 2 6 bitrdi 
 |-  ( A e. ( SubRing ` R ) -> ( a e. ( SubRing ` S ) <-> a e. ( ( SubRing ` R ) i^i ~P A ) ) )
8 7 eqrdv 
 |-  ( A e. ( SubRing ` R ) -> ( SubRing ` S ) = ( ( SubRing ` R ) i^i ~P A ) )