Metamath Proof Explorer


Theorem dfres2

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

Ref Expression
Assertion dfres2 ( 𝑅𝐴 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑥 𝑅 𝑦 ) }

Proof

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