Theorem brab2a

Description: The law of concretion for a binary relation. Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Hypotheses	brab2a.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
	brab2a.2	⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) }
Assertion	brab2a	⊢ ( 𝐴 𝑅 𝐵 ↔ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		brab2a.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
2		brab2a.2	⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) }
3		opabssxp	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) } ⊆ ( 𝐶 × 𝐷 )
4	2 3	eqsstri	⊢ 𝑅 ⊆ ( 𝐶 × 𝐷 )
5	4	brel	⊢ ( 𝐴 𝑅 𝐵 → ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) )
6		df-br	⊢ ( 𝐴 𝑅 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 )
7	2	eleq2i	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) } )
8	6 7	bitri	⊢ ( 𝐴 𝑅 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) } )
9	1	opelopab2a	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) } ↔ 𝜓 ) )
10	8 9	bitrid	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( 𝐴 𝑅 𝐵 ↔ 𝜓 ) )
11	5 10	biadanii	⊢ ( 𝐴 𝑅 𝐵 ↔ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) ∧ 𝜓 ) )