Theorem elrest

Description: The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009) (Revised by Mario Carneiro, 15-Dec-2013)

	Ref	Expression
Assertion	elrest	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∈ ( 𝐽 ↾t 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐽 𝐴 = ( 𝑥 ∩ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		restval	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐽 ↾t 𝐵 ) = ran ( 𝑥 ∈ 𝐽 ↦ ( 𝑥 ∩ 𝐵 ) ) )
2	1	eleq2d	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∈ ( 𝐽 ↾t 𝐵 ) ↔ 𝐴 ∈ ran ( 𝑥 ∈ 𝐽 ↦ ( 𝑥 ∩ 𝐵 ) ) ) )
3		eqid	⊢ ( 𝑥 ∈ 𝐽 ↦ ( 𝑥 ∩ 𝐵 ) ) = ( 𝑥 ∈ 𝐽 ↦ ( 𝑥 ∩ 𝐵 ) )
4		vex	⊢ 𝑥 ∈ V
5	4	inex1	⊢ ( 𝑥 ∩ 𝐵 ) ∈ V
6	3 5	elrnmpti	⊢ ( 𝐴 ∈ ran ( 𝑥 ∈ 𝐽 ↦ ( 𝑥 ∩ 𝐵 ) ) ↔ ∃ 𝑥 ∈ 𝐽 𝐴 = ( 𝑥 ∩ 𝐵 ) )
7	2 6	bitrdi	⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∈ ( 𝐽 ↾t 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐽 𝐴 = ( 𝑥 ∩ 𝐵 ) ) )