Theorem eqrel2

Description: Equality of relations. (Contributed by Peter Mazsa, 8-Mar-2019)

	Ref	Expression
Assertion	eqrel2	$$⊢\left(\mathrm{Rel}A\wedge \mathrm{Rel}B\right)\to \left(A=B\leftrightarrow \forall x\forall y\left(xAy\leftrightarrow xBy\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ssrel3	$$⊢\mathrm{Rel}A\to \left(A\subseteq B\leftrightarrow \forall x\forall y\left(xAy\to xBy\right)\right)$$
2		ssrel3	$$⊢\mathrm{Rel}B\to \left(B\subseteq A\leftrightarrow \forall x\forall y\left(xBy\to xAy\right)\right)$$
3	1 2	bi2anan9	$$⊢\left(\mathrm{Rel}A\wedge \mathrm{Rel}B\right)\to \left(\left(A\subseteq B\wedge B\subseteq A\right)\leftrightarrow \left(\forall x\forall y\left(xAy\to xBy\right)\wedge \forall x\forall y\left(xBy\to xAy\right)\right)\right)$$
4		eqss	$$⊢A=B\leftrightarrow \left(A\subseteq B\wedge B\subseteq A\right)$$
5		2albiim	$$⊢\forall x\forall y\left(xAy\leftrightarrow xBy\right)\leftrightarrow \left(\forall x\forall y\left(xAy\to xBy\right)\wedge \forall x\forall y\left(xBy\to xAy\right)\right)$$
6	3 4 5	3bitr4g	$$⊢\left(\mathrm{Rel}A\wedge \mathrm{Rel}B\right)\to \left(A=B\leftrightarrow \forall x\forall y\left(xAy\leftrightarrow xBy\right)\right)$$