Theorem nllytop 23641

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 23637	. 2 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))
2	1	simplbi 501	1 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  ∀wral 3078  ∃wrex 3088   ∩ cin 3903  𝒫 cpw 4561  {csn 4588  ‘cfv 6536  (class class class)co 7412   ↾_t crest 17479  Topctop 23061  neicnei 23265  𝑛-Locally cnlly 23633

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7415  df-nlly 23635

This theorem is used by:  nlly2i  23644  restnlly  23650  nllyrest  23654  nllyidm  23657  cldllycmp  23663  llycmpkgen  23720  txnlly  23805  txkgen  23820  xkococnlem  23827  xkococn  23828  cnmptkk  23851  xkofvcn  23852  cnmptk1p  23853  cnmptk2  23854  xkocnv  23982  xkohmeo  23983