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Theorem neii1 23129
Description: A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.)
Hypothesis
Ref Expression
neifval.1 𝑋 = 𝐽
Assertion
Ref Expression
neii1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁𝑋)

Proof of Theorem neii1
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . 3 𝑋 = 𝐽
21neiss2 23124 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
31isnei 23126 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))))
4 simpl 482 . . . 4 ((𝑁𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑁)) → 𝑁𝑋)
53, 4biimtrdi 253 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑁𝑋))
65impancom 451 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑆𝑋𝑁𝑋))
72, 6mpd 15 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wrex 3067  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:  neisspw  23130  neiss  23132  opnnei  23143  neiuni  23145  topssnei  23147  innei  23148  neissex  23150  iscnp4  23286  llycmpkgen2  23573  neitx  23630  flimopn  23998  flfnei  24014  fclsneii  24040  fcfnei  24058  cnextcn  24090  limcflf  25930  cvmlift2lem1  35286  neiin  36314  neibastop2  36343  cnneiima  48712
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