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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 23141. (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 22927 | . . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
3 | opnneiss 23141 | . . . 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 23133 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋) | |
7 | 6 | ex 412 | . 2 ⊢ (𝐽 ∈ Top → (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ⊆ 𝑋)) |
8 | 5, 7 | impbid 212 | 1 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 ∪ cuni 4911 ‘cfv 6562 Topctop 22914 neicnei 23120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-top 22915 df-nei 23121 |
This theorem is referenced by: neiuni 23145 neifil 23903 gneispa 44119 |
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