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Theorem neiss2 21712
Description: A set with a neighborhood is a subset of the base set of a topology. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.)
Hypothesis
Ref Expression
neifval.1 𝑋 = 𝐽
Assertion
Ref Expression
neiss2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)

Proof of Theorem neiss2
StepHypRef Expression
1 elfvdm 6694 . . . 4 (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ∈ dom (nei‘𝐽))
21adantl 485 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ dom (nei‘𝐽))
3 neifval.1 . . . . . . 7 𝑋 = 𝐽
43neif 21711 . . . . . 6 (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋)
54fndmd 6446 . . . . 5 (𝐽 ∈ Top → dom (nei‘𝐽) = 𝒫 𝑋)
65eleq2d 2901 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋))
76adantr 484 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋))
82, 7mpbid 235 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ 𝒫 𝑋)
98elpwid 4534 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wss 3920  𝒫 cpw 4523   cuni 4825  dom cdm 5543  cfv 6344  Topctop 21504  neicnei 21708
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 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  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 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-top 21505  df-nei 21709
This theorem is referenced by:  neii1  21717  neii2  21719  neiss  21720  ssnei2  21727  topssnei  21735  innei  21736  neitx  22218  cvmlift2lem12  32621  neiin  33740
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