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Theorem neiss2 20815
Description: A set with a neighborhood is a subset of the topology's base set. (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 6177 . . . 4 (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ∈ dom (nei‘𝐽))
21adantl 482 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ dom (nei‘𝐽))
3 neifval.1 . . . . . . 7 𝑋 = 𝐽
43neif 20814 . . . . . 6 (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋)
5 fndm 5948 . . . . . 6 ((nei‘𝐽) Fn 𝒫 𝑋 → dom (nei‘𝐽) = 𝒫 𝑋)
64, 5syl 17 . . . . 5 (𝐽 ∈ Top → dom (nei‘𝐽) = 𝒫 𝑋)
76eleq2d 2684 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋))
87adantr 481 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋))
92, 8mpbid 222 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ 𝒫 𝑋)
109elpwid 4141 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wss 3555  𝒫 cpw 4130   cuni 4402  dom cdm 5074   Fn wfn 5842  cfv 5847  Topctop 20617  neicnei 20811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-top 20621  df-nei 20812
This theorem is referenced by:  neii1  20820  neii2  20822  neiss  20823  ssnei2  20830  topssnei  20838  innei  20839  neitx  21320  cvmlift2lem12  31004  neiin  31969
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