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	$$⊢\mathrm{I}\cap \left(A\times A\right)\subseteq R\cap \left(A\times A\right)\leftrightarrow {\mathrm{I}\upharpoonright }_A\subseteq R$$

Proof

Step	Hyp	Ref	Expression
1		idinxpssinxp2	$$⊢\mathrm{I}\cap \left(A\times A\right)\subseteq R\cap \left(A\times A\right)\leftrightarrow \forall x\in AxRx$$
2		idrefALT	$$⊢{\mathrm{I}\upharpoonright }_A\subseteq R\leftrightarrow \forall x\in AxRx$$
3	1 2	bitr4i	$$⊢\mathrm{I}\cap \left(A\times A\right)\subseteq R\cap \left(A\times A\right)\leftrightarrow {\mathrm{I}\upharpoonright }_A\subseteq R$$