Theorem brab2d

Description: Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by Thierry Arnoux, 4-May-2025)

	Ref	Expression
Hypotheses	brab2d.1	⊢ ( 𝜑 → 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) } )
	brab2d.2	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	brab2d	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		brab2d.1	⊢ ( 𝜑 → 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) } )
2		brab2d.2	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( 𝜓 ↔ 𝜒 ) )
3		df-br	⊢ ( 𝐴 𝑅 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 )
4	1	eleq2d	⊢ ( 𝜑 → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) } ) )
5	3 4	bitrid	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) } ) )
6		elopab	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) } ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) ) )
7	5 6	bitrdi	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) ) ) )
8		eqcom	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ↔ ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ )
9		vex	⊢ 𝑥 ∈ V
10		vex	⊢ 𝑦 ∈ V
11	9 10	opth	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) )
12	8 11	sylbb1	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) )
13		eleq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈 ) )
14		eleq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉 ) )
15	13 14	bi2anan9	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ↔ ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ) )
16	15	biimpa	⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) )
17	12 16	sylan	⊢ ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) )
18	17	adantl	⊢ ( ( 𝜑 ∧ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ) ) → ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) )
19	18	adantrrr	⊢ ( ( 𝜑 ∧ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) ) ) → ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) )
20	19	ex	⊢ ( 𝜑 → ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) ) → ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ) )
21	20	exlimdvv	⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) ) → ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ) )
22	21	imp	⊢ ( ( 𝜑 ∧ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) ) ) → ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) )
23		simprl	⊢ ( ( 𝜑 ∧ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝜒 ) ) → ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) )
24		simprl	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ) → 𝐴 ∈ 𝑈 )
25		simprr	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ) → 𝐵 ∈ 𝑉 )
26	15	adantl	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ↔ ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ) )
27	26 2	anbi12d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) ↔ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝜒 ) ) )
28	27	adantlr	⊢ ( ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ) ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → ( ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) ↔ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝜒 ) ) )
29	24 25 28	copsex2dv	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) ) ↔ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝜒 ) ) )
30	22 23 29	bibiad	⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ( ( 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝜓 ) ) ↔ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝜒 ) ) )
31	7 30	bitrd	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝜒 ) ) )