Theorem nllytop 22976

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 22972	. 2 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))
2	1	simplbi 498	1 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070   ∩ cin 3947  𝒫 cpw 4602  {csn 4628  ‘cfv 6543  (class class class)co 7408   ↾_t crest 17365  Topctop 22394  neicnei 22600  𝑛-Locally cnlly 22968

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-nlly 22970

This theorem is referenced by:  nlly2i  22979  restnlly  22985  nllyrest  22989  nllyidm  22992  cldllycmp  22998  llycmpkgen  23055  txnlly  23140  txkgen  23155  xkococnlem  23162  xkococn  23163  cnmptkk  23186  xkofvcn  23187  cnmptk1p  23188  cnmptk2  23189  xkocnv  23317  xkohmeo  23318