ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  neiss2 Unicode version

Theorem neiss2 12857
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 12856 . . . . 5  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )
3 fnrel 5294 . . . . 5  |-  ( ( nei `  J )  Fn  ~P X  ->  Rel  ( nei `  J
) )
42, 3syl 14 . . . 4  |-  ( J  e.  Top  ->  Rel  ( nei `  J ) )
5 relelfvdm 5526 . . . 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 5295 . . . . . 6  |-  ( ( nei `  J )  Fn  ~P X  ->  dom  ( nei `  J
)  =  ~P X
)
82, 7syl 14 . . . . 5  |-  ( J  e.  Top  ->  dom  ( nei `  J )  =  ~P X )
98eleq2d 2240 . . . 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 3575 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 1348    e. wcel 2141    C_ wss 3121   ~Pcpw 3564   U.cuni 3794   dom cdm 4609   Rel wrel 4614    Fn wfn 5191   ` cfv 5196   Topctop 12710   neicnei 12853
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-top 12711  df-nei 12854
This theorem is referenced by:  neii1  12862  neii2  12864  neiss  12865  ssnei2  12872  topssnei  12877  innei  12878  neitx  12983
  Copyright terms: Public domain W3C validator