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Metamath Proof Explorer

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 ) }