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Metamath Proof Explorer

Theorem dmres

Description: The domain of a restriction. Exercise 14 of TakeutiZaring p. 25. (Contributed by NM, 1-Aug-1994)

Ref Expression
Assertion dmres dom ( 𝐴 ↾ 𝐵 ) = ( 𝐵 ∩ dom 𝐴 )

Proof

Step Hyp Ref Expression
1 vex ⊢ 𝑥 ∈ V
2 1 eldm2 ⊢ ( 𝑥 ∈ dom ( 𝐴 ↾ 𝐵 ) ↔ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 ↾ 𝐵 ) )
3 19.42v ⊢ ( ∃ 𝑦 ( 𝑥 ∈ 𝐵 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) )
4 vex ⊢ 𝑦 ∈ V
5 4 opelresi ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 ↾ 𝐵 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) )
6 5 exbii ⊢ ( ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 ↾ 𝐵 ) ↔ ∃ 𝑦 ( 𝑥 ∈ 𝐵 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) )
7 1 eldm2 ⊢ ( 𝑥 ∈ dom 𝐴 ↔ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 )
8 7 anbi2i ⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐴 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) )
9 3 6 8 3bitr4i ⊢ ( ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ ( 𝐴 ↾ 𝐵 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐴 ) )
10 2 9 bitr2i ⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐴 ) ↔ 𝑥 ∈ dom ( 𝐴 ↾ 𝐵 ) )
11 10 ineqri ⊢ ( 𝐵 ∩ dom 𝐴 ) = dom ( 𝐴 ↾ 𝐵 )
12 11 eqcomi ⊢ dom ( 𝐴 ↾ 𝐵 ) = ( 𝐵 ∩ dom 𝐴 )