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Theorem neisspw 12559
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 12558 . . . 4 ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘𝑆)) → 𝑣𝑋)
3 velpw 3550 . . . 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 3134 1 (𝐽 ∈ Top → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wcel 2128  wss 3102  𝒫 cpw 3543   cuni 3772  cfv 5170  Topctop 12406  neicnei 12549
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-top 12407  df-nei 12550
This theorem is referenced by: (None)
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