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Theorem neisspw 12788
Description: The neighborhoods of any set are subsets of the base set. (Contributed by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
neifval.1 𝑋 = 𝐽
Assertion
Ref Expression
neisspw (𝐽 ∈ Top → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)

Proof of Theorem neisspw
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . . 5 𝑋 = 𝐽
21neii1 12787 . . . 4 ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘𝑆)) → 𝑣𝑋)
3 velpw 3566 . . . 4 (𝑣 ∈ 𝒫 𝑋𝑣𝑋)
42, 3sylibr 133 . . 3 ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘𝑆)) → 𝑣 ∈ 𝒫 𝑋)
54ex 114 . 2 (𝐽 ∈ Top → (𝑣 ∈ ((nei‘𝐽)‘𝑆) → 𝑣 ∈ 𝒫 𝑋))
65ssrdv 3148 1 (𝐽 ∈ Top → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1343  wcel 2136  wss 3116  𝒫 cpw 3559   cuni 3789  cfv 5188  Topctop 12635  neicnei 12778
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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-top 12636  df-nei 12779
This theorem is referenced by: (None)
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