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Theorem neiss2 12320
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 neifval.1 . . . . . 6 𝑋 = 𝐽
21neif 12319 . . . . 5 (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋)
3 fnrel 5221 . . . . 5 ((nei‘𝐽) Fn 𝒫 𝑋 → Rel (nei‘𝐽))
42, 3syl 14 . . . 4 (𝐽 ∈ Top → Rel (nei‘𝐽))
5 relelfvdm 5453 . . . 4 ((Rel (nei‘𝐽) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ dom (nei‘𝐽))
64, 5sylan 281 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ dom (nei‘𝐽))
7 fndm 5222 . . . . . 6 ((nei‘𝐽) Fn 𝒫 𝑋 → dom (nei‘𝐽) = 𝒫 𝑋)
82, 7syl 14 . . . . 5 (𝐽 ∈ Top → dom (nei‘𝐽) = 𝒫 𝑋)
98eleq2d 2209 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋))
109adantr 274 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋))
116, 10mpbid 146 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ 𝒫 𝑋)
1211elpwid 3521 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wss 3071  𝒫 cpw 3510   cuni 3736  dom cdm 4539  Rel wrel 4544   Fn wfn 5118  cfv 5123  Topctop 12173  neicnei 12316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-top 12174  df-nei 12317
This theorem is referenced by:  neii1  12325  neii2  12327  neiss  12328  ssnei2  12335  topssnei  12340  innei  12341  neitx  12446
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