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Theorem tpnei 21146
Description: The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 21143. (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 20931 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 opnneiss 21143 . . . 4 ((𝐽 ∈ Top ∧ 𝑋𝐽𝑆𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
433exp 1112 . . 3 (𝐽 ∈ Top → (𝑋𝐽 → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆))))
52, 4mpd 15 . 2 (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))
6 ssnei 21135 . . 3 ((𝐽 ∈ Top ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
76ex 397 . 2 (𝐽 ∈ Top → (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑆𝑋))
85, 7impbid 202 1 (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  wss 3723   cuni 4574  cfv 6031  Topctop 20918  neicnei 21122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-top 20919  df-nei 21123
This theorem is referenced by:  neiuni  21147  neifil  21904  gneispa  38954
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