Theorem nllytop 23452

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 23448	. 2 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))
2	1	simplbi 496	1 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3889  𝒫 cpw 4542  {csn 4568  ‘cfv 6494  (class class class)co 7362   ↾_t crest 17378  Topctop 22872  neicnei 23076  𝑛-Locally cnlly 23444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6450  df-fv 6502  df-ov 7365  df-nlly 23446

This theorem is referenced by:  nlly2i  23455  restnlly  23461  nllyrest  23465  nllyidm  23468  cldllycmp  23474  llycmpkgen  23531  txnlly  23616  txkgen  23631  xkococnlem  23638  xkococn  23639  cnmptkk  23662  xkofvcn  23663  cnmptk1p  23664  cnmptk2  23665  xkocnv  23793  xkohmeo  23794