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Theorem llytop 21215
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 21211 . 2 (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
21simplbi 476 1 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2908  wrex 2909  cin 3559  𝒫 cpw 4136  (class class class)co 6615  t crest 16021  Topctop 20638  Locally clly 21207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-lly 21209
This theorem is referenced by:  llynlly  21220  islly2  21227  llyrest  21228  llyidm  21231  nllyidm  21232  toplly  21233  lly1stc  21239  txlly  21379
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