Theorem nllytop 23753

Description: A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
nllytop	⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)

Proof of Theorem nllytop

Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isnlly 23749	. 2 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))
2	1	simplbi 502	1 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∩ cin 3897  𝒫 cpw 4556  {csn 4583  ‘cfv 6527  (class class class)co 7408   ↾_t crest 17552  Topctop 23172  neicnei 23376  𝑛-Locally cnlly 23745

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-ext 2732

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-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6483  df-fv 6535  df-ov 7411  df-nlly 23747

This theorem is used by:  nlly2i  23756  restnlly  23762  nllyrest  23766  nllyidm  23769  cldllycmp  23775  llycmpkgen  23832  txnlly  23917  txkgen  23932  xkococnlem  23939  xkococn  23940  cnmptkk  23963  xkofvcn  23964  cnmptk1p  23965  cnmptk2  23966  xkocnv  24094  xkohmeo  24095