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Theorem llytop 23496
Description: A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llytop (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)

Proof of Theorem llytop
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 islly 23492 . 2 (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
21simplbi 497 1 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wral 3059  wrex 3068  cin 3962  𝒫 cpw 4605  (class class class)co 7431  t crest 17467  Topctop 22915  Locally clly 23488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-lly 23490
This theorem is referenced by:  llynlly  23501  islly2  23508  llyrest  23509  llyidm  23512  nllyidm  23513  toplly  23514  lly1stc  23520  txlly  23660
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