MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neisspw Structured version   Visualization version   GIF version

Theorem neisspw 22962
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 = βˆͺ 𝐽
Assertion
Ref Expression
neisspw (𝐽 ∈ Top β†’ ((neiβ€˜π½)β€˜π‘†) βŠ† 𝒫 𝑋)

Proof of Theorem neisspw
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . . 5 𝑋 = βˆͺ 𝐽
21neii1 22961 . . . 4 ((𝐽 ∈ Top ∧ 𝑣 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑣 βŠ† 𝑋)
3 velpw 4602 . . . 4 (𝑣 ∈ 𝒫 𝑋 ↔ 𝑣 βŠ† 𝑋)
42, 3sylibr 233 . . 3 ((𝐽 ∈ Top ∧ 𝑣 ∈ ((neiβ€˜π½)β€˜π‘†)) β†’ 𝑣 ∈ 𝒫 𝑋)
54ex 412 . 2 (𝐽 ∈ Top β†’ (𝑣 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝑣 ∈ 𝒫 𝑋))
65ssrdv 3983 1 (𝐽 ∈ Top β†’ ((neiβ€˜π½)β€˜π‘†) βŠ† 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  π’« cpw 4597  βˆͺ cuni 4902  β€˜cfv 6536  Topctop 22746  neicnei 22952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 22747  df-nei 22953
This theorem is referenced by:  hausflim  23836  flimclslem  23839  fclsfnflim  23882
  Copyright terms: Public domain W3C validator