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Theorem neiss2 12154
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 12153 . . . . 5  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )
3 fnrel 5179 . . . . 5  |-  ( ( nei `  J )  Fn  ~P X  ->  Rel  ( nei `  J
) )
42, 3syl 14 . . . 4  |-  ( J  e.  Top  ->  Rel  ( nei `  J ) )
5 relelfvdm 5407 . . . 4  |-  ( ( Rel  ( nei `  J
)  /\  N  e.  ( ( nei `  J
) `  S )
)  ->  S  e.  dom  ( nei `  J
) )
64, 5sylan 279 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  e.  dom  ( nei `  J ) )
7 fndm 5180 . . . . . 6  |-  ( ( nei `  J )  Fn  ~P X  ->  dom  ( nei `  J
)  =  ~P X
)
82, 7syl 14 . . . . 5  |-  ( J  e.  Top  ->  dom  ( nei `  J )  =  ~P X )
98eleq2d 2184 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  dom  ( nei `  J )  <->  S  e.  ~P X ) )
109adantr 272 . . 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 3487 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 1314    e. wcel 1463    C_ wss 3037   ~Pcpw 3476   U.cuni 3702   dom cdm 4499   Rel wrel 4504    Fn wfn 5076   ` cfv 5081   Topctop 12007   neicnei 12150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-top 12008  df-nei 12151
This theorem is referenced by:  neii1  12159  neii2  12161  neiss  12162  ssnei2  12169  topssnei  12174  innei  12175  neitx  12279
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