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Theorem tpnei 21735
 Description: The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 21732. (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 21520 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 opnneiss 21732 . . . 4 ((𝐽 ∈ Top ∧ 𝑋𝐽𝑆𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
433exp 1116 . . 3 (𝐽 ∈ Top → (𝑋𝐽 → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆))))
52, 4mpd 15 . 2 (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))
6 ssnei 21724 . . 3 ((𝐽 ∈ Top ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
76ex 416 . 2 (𝐽 ∈ Top → (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑆𝑋))
85, 7impbid 215 1 (𝐽 ∈ Top → (𝑆𝑋𝑋 ∈ ((nei‘𝐽)‘𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2115   ⊆ wss 3919  ∪ cuni 4824  ‘cfv 6345  Topctop 21507  neicnei 21711 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-top 21508  df-nei 21712 This theorem is referenced by:  neiuni  21736  neifil  22494  gneispa  40780
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