Theorem llytop 22531

Description: A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
llytop	⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)

Proof of Theorem llytop

Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		islly 22527	. 2 ⊢ (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
2	1	simplbi 497	1 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∩ cin 3882  𝒫 cpw 4530  (class class class)co 7255   ↾_t crest 17048  Topctop 21950  Locally clly 22523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-lly 22525

This theorem is referenced by:  llynlly  22536  islly2  22543  llyrest  22544  llyidm  22547  nllyidm  22548  toplly  22549  lly1stc  22555  txlly  22695