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Theorem neiss2 16838
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  = 
U. J
Assertion
Ref Expression
neiss2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )

Proof of Theorem neiss2
StepHypRef Expression
1 elfvdm 5554 . . . 4  |-  ( N  e.  ( ( nei `  J ) `  S
)  ->  S  e.  dom  ( nei `  J
) )
21adantl 452 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  e.  dom  ( nei `  J ) )
3 neifval.1 . . . . . . 7  |-  X  = 
U. J
43neif 16837 . . . . . 6  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )
5 fndm 5343 . . . . . 6  |-  ( ( nei `  J )  Fn  ~P X  ->  dom  ( nei `  J
)  =  ~P X
)
64, 5syl 15 . . . . 5  |-  ( J  e.  Top  ->  dom  ( nei `  J )  =  ~P X )
76eleq2d 2350 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  dom  ( nei `  J )  <->  S  e.  ~P X ) )
87adantr 451 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  -> 
( S  e.  dom  ( nei `  J )  <-> 
S  e.  ~P X
) )
92, 8mpbid 201 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  e.  ~P X
)
10 uniexg 4517 . . . . 5  |-  ( J  e.  Top  ->  U. J  e.  _V )
113, 10syl5eqel 2367 . . . 4  |-  ( J  e.  Top  ->  X  e.  _V )
12 elpw2g 4174 . . . 4  |-  ( X  e.  _V  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
1311, 12syl 15 . . 3  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
1413adantr 451 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  -> 
( S  e.  ~P X 
<->  S  C_  X )
)
159, 14mpbid 201 1  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   dom cdm 4689    Fn wfn 5250   ` cfv 5255   Topctop 16631   neicnei 16834
This theorem is referenced by:  neii1  16843  neii2  16845  neiss  16846  ssnei2  16853  topssnei  16861  innei  16862  cvmlift2lem12  23845  neiin  26250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-nei 16835
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