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Theorem neiss2 12348
Description: A set with a neighborhood is a subset of the base set of a topology. (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 neifval.1 . . . . . 6  |-  X  = 
U. J
21neif 12347 . . . . 5  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )
3 fnrel 5228 . . . . 5  |-  ( ( nei `  J )  Fn  ~P X  ->  Rel  ( nei `  J
) )
42, 3syl 14 . . . 4  |-  ( J  e.  Top  ->  Rel  ( nei `  J ) )
5 relelfvdm 5460 . . . 4  |-  ( ( Rel  ( nei `  J
)  /\  N  e.  ( ( nei `  J
) `  S )
)  ->  S  e.  dom  ( nei `  J
) )
64, 5sylan 281 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  e.  dom  ( nei `  J ) )
7 fndm 5229 . . . . . 6  |-  ( ( nei `  J )  Fn  ~P X  ->  dom  ( nei `  J
)  =  ~P X
)
82, 7syl 14 . . . . 5  |-  ( J  e.  Top  ->  dom  ( nei `  J )  =  ~P X )
98eleq2d 2210 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  dom  ( nei `  J )  <->  S  e.  ~P X ) )
109adantr 274 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  -> 
( S  e.  dom  ( nei `  J )  <-> 
S  e.  ~P X
) )
116, 10mpbid 146 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  e.  ~P X
)
1211elpwid 3525 1  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481    C_ wss 3075   ~Pcpw 3514   U.cuni 3743   dom cdm 4546   Rel wrel 4551    Fn wfn 5125   ` cfv 5130   Topctop 12201   neicnei 12344
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-top 12202  df-nei 12345
This theorem is referenced by:  neii1  12353  neii2  12355  neiss  12356  ssnei2  12363  topssnei  12368  innei  12369  neitx  12474
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