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Theorem tpnei 22282
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.)
Hypothesis
Ref Expression
tpnei.1 𝑋 = 𝐽
Assertion
Ref Expression
tpnei (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))

Proof of Theorem tpnei
StepHypRef Expression
1 tpnei.1 . . . 4 𝑋 = 𝐽
21topopn 22065 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 opnneiss 22279 . . . 4 ((𝐽 ∈ Top ∧ 𝑋𝐽𝑆𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
433exp 1118 . . 3 (𝐽 ∈ Top → (𝑋𝐽 → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆))))
52, 4mpd 15 . 2 (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))
6 ssnei 22271 . . 3 ((𝐽 ∈ Top ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
76ex 413 . 2 (𝐽 ∈ Top → (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑆𝑋))
85, 7impbid 211 1 (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((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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