Theorem ecxpid

Description: The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024)

	Ref	Expression
Assertion	ecxpid	$$⊢X\in A\to \left[X\right]\left(A\times A\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2		elecg	$$⊢\left(x\in \mathrm{V}\wedge X\in A\right)\to \left(x\in \left[X\right]\left(A\times A\right)\leftrightarrow X\left(A\times A\right)x\right)$$
3	1 2	mpan	$$⊢X\in A\to \left(x\in \left[X\right]\left(A\times A\right)\leftrightarrow X\left(A\times A\right)x\right)$$
4		brxp	$$⊢X\left(A\times A\right)x\leftrightarrow \left(X\in A\wedge x\in A\right)$$
5	4	baib	$$⊢X\in A\to \left(X\left(A\times A\right)x\leftrightarrow x\in A\right)$$
6	3 5	bitrd	$$⊢X\in A\to \left(x\in \left[X\right]\left(A\times A\right)\leftrightarrow x\in A\right)$$
7	6	eqrdv	$$⊢X\in A\to \left[X\right]\left(A\times A\right)=A$$