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	$⊢ (J \in Top \land N \in nei (J) (S)) \to S \subseteq X$

Proof

Step	Hyp	Ref	Expression
1		neifval.1	$⊢ X = ⋃ J$
2		elfvdm	$⊢ N \in nei (J) (S) \to S \in dom nei (J)$
3	2	adantl	$⊢ (J \in Top \land N \in nei (J) (S)) \to S \in dom nei (J)$
4	1	neif	$⊢ J \in Top \to nei (J) Fn 𝒫 X$
5	4	fndmd	$⊢ J \in Top \to dom nei (J) = 𝒫 X$
6	5	eleq2d	$⊢ J \in Top \to (S \in dom nei (J) \leftrightarrow S \in 𝒫 X)$
7	6	adantr	$⊢ (J \in Top \land N \in nei (J) (S)) \to (S \in dom nei (J) \leftrightarrow S \in 𝒫 X)$
8	3 7	mpbid	$⊢ (J \in Top \land N \in nei (J) (S)) \to S \in 𝒫 X$
9	8	elpwid	$⊢ (J \in Top \land N \in nei (J) (S)) \to S \subseteq X$

Metamath Proof Explorer

Theorem neiss2

Proof