Theorem elopabran

Description: Membership in an ordered-pair class abstraction defined by a restricted binary relation. (Contributed by AV, 16-Feb-2021)

	Ref	Expression
Assertion	elopabran	$$⊢A\in \left\{⟨x,y⟩|\left(xRy\wedge \psi \right)\right\}\to A\in R$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(xRy\wedge \psi \right)\to xRy$$
2	1	ssopab2i	$$⊢\left\{⟨x,y⟩|\left(xRy\wedge \psi \right)\right\}\subseteq \left\{⟨x,y⟩|xRy\right\}$$
3	2	sseli	$$⊢A\in \left\{⟨x,y⟩|\left(xRy\wedge \psi \right)\right\}\to A\in \left\{⟨x,y⟩|xRy\right\}$$
4		elopabr	$$⊢A\in \left\{⟨x,y⟩|xRy\right\}\to A\in R$$
5	3 4	syl	$$⊢A\in \left\{⟨x,y⟩|\left(xRy\wedge \psi \right)\right\}\to A\in R$$