Theorem dfres2

Description: Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013)

	Ref	Expression
Assertion	dfres2	⊢ ( 𝑅 ↾ 𝐴 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝑦 ) }

Proof

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