Theorem br1cnvinxp

Description: Binary relation on the converse of an intersection with a Cartesian product. (Contributed by Peter Mazsa, 27-Jul-2019)

	Ref	Expression
Assertion	br1cnvinxp	$$⊢C{\left(R\cap \left(A\times B\right)\right)}^{-1}D\leftrightarrow \left(\left(C\in B\wedge D\in A\right)\wedge DRC\right)$$

Proof

Step	Hyp	Ref	Expression
1		relinxp	$$⊢\mathrm{Rel}\left(R\cap \left(A\times B\right)\right)$$
2	1	relbrcnv	$$⊢C{\left(R\cap \left(A\times B\right)\right)}^{-1}D\leftrightarrow D\left(R\cap \left(A\times B\right)\right)C$$
3		brinxp2	$$⊢D\left(R\cap \left(A\times B\right)\right)C\leftrightarrow \left(\left(D\in A\wedge C\in B\right)\wedge DRC\right)$$
4		ancom	$$⊢\left(D\in A\wedge C\in B\right)\leftrightarrow \left(C\in B\wedge D\in A\right)$$
5	4	anbi1i	$$⊢\left(\left(D\in A\wedge C\in B\right)\wedge DRC\right)\leftrightarrow \left(\left(C\in B\wedge D\in A\right)\wedge DRC\right)$$
6	2 3 5	3bitri	$$⊢C{\left(R\cap \left(A\times B\right)\right)}^{-1}D\leftrightarrow \left(\left(C\in B\wedge D\in A\right)\wedge DRC\right)$$