Theorem inxprnres

Description: Restriction of a class as a class of ordered pairs. (Contributed by Peter Mazsa, 2-Jan-2019)

	Ref	Expression
Assertion	inxprnres	⊢ ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) }

Proof

Step	Hyp	Ref	Expression
1		relinxp	⊢ Rel ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) )
2		relopabv	⊢ Rel { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) }
3		eleq1w	⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴 ) )
4		breq1	⊢ ( 𝑥 = 𝑧 → ( 𝑥 𝑅 𝑦 ↔ 𝑧 𝑅 𝑦 ) )
5	3 4	anbi12d	⊢ ( 𝑥 = 𝑧 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) ↔ ( 𝑧 ∈ 𝐴 ∧ 𝑧 𝑅 𝑦 ) ) )
6		breq2	⊢ ( 𝑦 = 𝑤 → ( 𝑧 𝑅 𝑦 ↔ 𝑧 𝑅 𝑤 ) )
7	6	anbi2d	⊢ ( 𝑦 = 𝑤 → ( ( 𝑧 ∈ 𝐴 ∧ 𝑧 𝑅 𝑦 ) ↔ ( 𝑧 ∈ 𝐴 ∧ 𝑧 𝑅 𝑤 ) ) )
8	5 7	opelopabg	⊢ ( ( 𝑧 ∈ V ∧ 𝑤 ∈ V ) → ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) } ↔ ( 𝑧 ∈ 𝐴 ∧ 𝑧 𝑅 𝑤 ) ) )
9	8	el2v	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) } ↔ ( 𝑧 ∈ 𝐴 ∧ 𝑧 𝑅 𝑤 ) )
10		brinxprnres	⊢ ( 𝑤 ∈ V → ( 𝑧 ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) 𝑤 ↔ ( 𝑧 ∈ 𝐴 ∧ 𝑧 𝑅 𝑤 ) ) )
11	10	elv	⊢ ( 𝑧 ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) 𝑤 ↔ ( 𝑧 ∈ 𝐴 ∧ 𝑧 𝑅 𝑤 ) )
12		df-br	⊢ ( 𝑧 ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) 𝑤 ↔ ⟨ 𝑧 , 𝑤 ⟩ ∈ ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) )
13	9 11 12	3bitr2ri	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) ↔ ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) } )
14	1 2 13	eqrelriiv	⊢ ( 𝑅 ∩ ( 𝐴 × ran ( 𝑅 ↾ 𝐴 ) ) ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) }