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Theorem neiss2 16854
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 5570 . . . 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 16853 . . . . . 6  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )
5 fndm 5359 . . . . . 6  |-  ( ( nei `  J )  Fn  ~P X  ->  dom  ( nei `  J
)  =  ~P X
)
64, 5syl 15 . . . . 5  |-  ( J  e.  Top  ->  dom  ( nei `  J )  =  ~P X )
76eleq2d 2363 . . . 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 4533 . . . . 5  |-  ( J  e.  Top  ->  U. J  e.  _V )
113, 10syl5eqel 2380 . . . 4  |-  ( J  e.  Top  ->  X  e.  _V )
12 elpw2g 4190 . . . 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 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   dom cdm 4705    Fn wfn 5266   ` cfv 5271   Topctop 16647   neicnei 16850
This theorem is referenced by:  neii1  16859  neii2  16861  neiss  16862  ssnei2  16869  topssnei  16877  innei  16878  cvmlift2lem12  23860  neiin  26353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-nei 16851
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