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Theorem neii1 23114
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 23109 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
31isnei 23111 . . . 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 1540  wcel 2108  wrex 3070  wss 3951   cuni 4907  cfv 6561  Topctop 22899  neicnei 23105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-nei 23106
This theorem is referenced by:  neisspw  23115  neiss  23117  opnnei  23128  neiuni  23130  topssnei  23132  innei  23133  neissex  23135  iscnp4  23271  llycmpkgen2  23558  neitx  23615  flimopn  23983  flfnei  23999  fclsneii  24025  fcfnei  24043  cnextcn  24075  limcflf  25916  cvmlift2lem1  35307  neiin  36333  neibastop2  36362  cnneiima  48814
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