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Mirrors > Home > MPE Home > Th. List > llytop | Structured version Visualization version GIF version |
Description: A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
Ref | Expression |
---|---|
llytop | ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islly 23492 | . 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) |
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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