Theorem nllytop 23700

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 23696	. 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 3078  ∃wrex 3088   ∩ cin 3901  𝒫 cpw 4560  {csn 4587  ‘cfv 6537  (class class class)co 7416   ↾_t crest 17509  Topctop 23119  neicnei 23323  𝑛-Locally cnlly 23692

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 2734

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7419  df-nlly 23694

This theorem is used by:  nlly2i  23703  restnlly  23709  nllyrest  23713  nllyidm  23716  cldllycmp  23722  llycmpkgen  23779  txnlly  23864  txkgen  23879  xkococnlem  23886  xkococn  23887  cnmptkk  23910  xkofvcn  23911  cnmptk1p  23912  cnmptk2  23913  xkocnv  24041  xkohmeo  24042