Theorem idinxpssinxp3

Description: Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 16-Mar-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	idinxpssinxp3	|- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> ( _I |` A ) C_ R )

Proof

Step	Hyp	Ref	Expression
1		idinxpssinxp2	|- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> A. x e. A x R x )
2		idrefALT	|- ( ( _I |` A ) C_ R <-> A. x e. A x R x )
3	1 2	bitr4i	|- ( ( _I i^i ( A X. A ) ) C_ ( R i^i ( A X. A ) ) <-> ( _I |` A ) C_ R )