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Mirrors > Home > MPE Home > Th. List > tpnei | Structured version Visualization version GIF version |
Description: The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 22279. (Contributed by FL, 2-Oct-2006.) |
Ref | Expression |
---|---|
tpnei.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
tpnei | ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpnei.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | topopn 22065 | . . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
3 | opnneiss 22279 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆)) | |
4 | 3 | 3exp 1118 | . . 3 ⊢ (𝐽 ∈ Top → (𝑋 ∈ 𝐽 → (𝑆 ⊆ 𝑋 → 𝑋 ∈ ((nei‘𝐽)‘𝑆)))) |
5 | 2, 4 | mpd 15 | . 2 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 → 𝑋 ∈ ((nei‘𝐽)‘𝑆))) |
6 | ssnei 22271 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋) | |
7 | 6 | ex 413 | . 2 ⊢ (𝐽 ∈ Top → (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ⊆ 𝑋)) |
8 | 5, 7 | impbid 211 | 1 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 ⊆ wss 3886 ∪ cuni 4839 ‘cfv 6426 Topctop 22052 neicnei 22258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-top 22053 df-nei 22259 |
This theorem is referenced by: neiuni 22283 neifil 23041 gneispa 41721 |
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