Theorem neiss2

Description: A set with a neighborhood is a subset of the base set of a topology. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007)

	Ref	Expression
Hypothesis	neifval.1	⊢ 𝑋 = ∪ 𝐽
Assertion	neiss2	⊢ ( ( 𝐽 ∈ Top ∧ 𝑁 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ) → 𝑆 ⊆ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		neifval.1	⊢ 𝑋 = ∪ 𝐽
2		elfvdm	⊢ ( 𝑁 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) → 𝑆 ∈ dom ( nei ‘ 𝐽 ) )
3	2	adantl	⊢ ( ( 𝐽 ∈ Top ∧ 𝑁 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ) → 𝑆 ∈ dom ( nei ‘ 𝐽 ) )
4	1	neif	⊢ ( 𝐽 ∈ Top → ( nei ‘ 𝐽 ) Fn 𝒫 𝑋 )
5	4	fndmd	⊢ ( 𝐽 ∈ Top → dom ( nei ‘ 𝐽 ) = 𝒫 𝑋 )
6	5	eleq2d	⊢ ( 𝐽 ∈ Top → ( 𝑆 ∈ dom ( nei ‘ 𝐽 ) ↔ 𝑆 ∈ 𝒫 𝑋 ) )
7	6	adantr	⊢ ( ( 𝐽 ∈ Top ∧ 𝑁 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ) → ( 𝑆 ∈ dom ( nei ‘ 𝐽 ) ↔ 𝑆 ∈ 𝒫 𝑋 ) )
8	3 7	mpbid	⊢ ( ( 𝐽 ∈ Top ∧ 𝑁 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ) → 𝑆 ∈ 𝒫 𝑋 )
9	8	elpwid	⊢ ( ( 𝐽 ∈ Top ∧ 𝑁 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝑆 ) ) → 𝑆 ⊆ 𝑋 )