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 )