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 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 𝑅 𝑦 ∧ 𝜓 ) } → 𝐴 ∈ 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝑥 𝑅 𝑦 ∧ 𝜓 ) → 𝑥 𝑅 𝑦 )
2	1	ssopab2i	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 𝑅 𝑦 ∧ 𝜓 ) } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 }
3		opabss	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 } ⊆ 𝑅
4	2 3	sstri	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 𝑅 𝑦 ∧ 𝜓 ) } ⊆ 𝑅
5	4	sseli	⊢ ( 𝐴 ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 𝑅 𝑦 ∧ 𝜓 ) } → 𝐴 ∈ 𝑅 )