Metamath Proof Explorer

Theorem idinxpssinxp4

Description: Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (see also idinxpssinxp2 ). (Contributed by Peter Mazsa, 8-Mar-2019)

Ref Expression
Assertion idinxpssinxp4 
|- ( A. x e. A A. y e. A ( x = y -> x R y ) <-> A. x e. A x R x )

Proof

Step Hyp Ref Expression
1 idinxpssinxp 
 |-  ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A A. y e. A ( x = y -> x R y ) )
2 idinxpssinxp2 
 |-  ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A x R x )
3 1 2 bitr3i 
 |-  ( A. x e. A A. y e. A ( x = y -> x R y ) <-> A. x e. A x R x )