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Theorem nllytop 22977
Description: A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllytop (𝐽 ∈ 𝑛-Locally 𝐴 β†’ 𝐽 ∈ Top)

Proof of Theorem nllytop
Dummy variables 𝑒 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnlly 22973 . 2 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴))
21simplbi 499 1 (𝐽 ∈ 𝑛-Locally 𝐴 β†’ 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   ∩ cin 3948  π’« cpw 4603  {csn 4629  β€˜cfv 6544  (class class class)co 7409   β†Ύt crest 17366  Topctop 22395  neicnei 22601  π‘›-Locally cnlly 22969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-nlly 22971
This theorem is referenced by:  nlly2i  22980  restnlly  22986  nllyrest  22990  nllyidm  22993  cldllycmp  22999  llycmpkgen  23056  txnlly  23141  txkgen  23156  xkococnlem  23163  xkococn  23164  cnmptkk  23187  xkofvcn  23188  cnmptk1p  23189  cnmptk2  23190  xkocnv  23318  xkohmeo  23319
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