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Metamath Proof Explorer

Theorem idinxpssinxp2

Description: Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 4-Mar-2019) (Proof modification is discouraged.)

Ref Expression
Assertion idinxpssinxp2 
|- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A x R x )

Proof

Step Hyp Ref Expression
1 idinxpresid 
 |-  ( _I i^i ( A X. A ) ) = ( _I |` A )
2 1 sseq1i 
 |-  ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> ( _I |` A ) C_ ( R i^i ( A X. A ) ) )
3 idrefALT 
 |-  ( ( _I |` A ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A x ( R i^i ( A X. A ) ) x )
4 brinxp2 
 |-  ( x ( R i^i ( A X. A ) ) x <-> ( ( x e. A /\ x e. A ) /\ x R x ) )
5 pm4.24 
 |-  ( x e. A <-> ( x e. A /\ x e. A ) )
6 5 anbi1i 
 |-  ( ( x e. A /\ x R x ) <-> ( ( x e. A /\ x e. A ) /\ x R x ) )
7 4 6 bitr4i 
 |-  ( x ( R i^i ( A X. A ) ) x <-> ( x e. A /\ x R x ) )
8 7 ralbii 
 |-  ( A. x e. A x ( R i^i ( A X. A ) ) x <-> A. x e. A ( x e. A /\ x R x ) )
9 2 3 8 3bitri 
 |-  ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A ( x e. A /\ x R x ) )
10 ralanid 
 |-  ( A. x e. A ( x e. A /\ x R x ) <-> A. x e. A x R x )
11 9 10 bitri 
 |-  ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A x R x )