Theorem neiss2 23326

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.)

Hypothesis

Ref	Expression
neifval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
neiss2	⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)

Proof of Theorem neiss2

Step	Hyp	Ref	Expression
1		elfvdm 6912	. . . 4 ⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ∈ dom (nei‘𝐽))
2	1	adantl 487	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ dom (nei‘𝐽))
3		neifval.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
4	3	neif 23325	. . . . . 6 ⊢ (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋)
5	4	fndmd 6637	. . . . 5 ⊢ (𝐽 ∈ Top → dom (nei‘𝐽) = 𝒫 𝑋)
6	5	eleq2d 2846	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋))
7	6	adantr 486	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋))
8	2, 7	mpbid 235	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ 𝒫 𝑋)
9	8	elpwid 4566	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  dom cdm 5655  ‘cfv 6533  Topctop 23118  neicnei 23322

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-top 23119  df-nei 23323

This theorem is used by:  neii1  23331  neii2  23333  neiss  23334  ssnei2  23341  topssnei  23349  innei  23350  neitx  23833  cvmlift2lem12  35893  neiin  36951  cnneiima  49843