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| Description: A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) | 
| Ref | Expression | 
|---|---|
| nllytop | ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isnlly 23478 | . 2 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 ∩ cin 3949 𝒫 cpw 4599 {csn 4625 ‘cfv 6560 (class class class)co 7432 ↾t crest 17466 Topctop 22900 neicnei 23106 𝑛-Locally cnlly 23474 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-nlly 23476 | 
| This theorem is referenced by: nlly2i 23485 restnlly 23491 nllyrest 23495 nllyidm 23498 cldllycmp 23504 llycmpkgen 23561 txnlly 23646 txkgen 23661 xkococnlem 23668 xkococn 23669 cnmptkk 23692 xkofvcn 23693 cnmptk1p 23694 cnmptk2 23695 xkocnv 23823 xkohmeo 23824 | 
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