MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neiss2 Unicode version

Theorem neiss2 17081
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 5690 . . . 4  |-  ( N  e.  ( ( nei `  J ) `  S
)  ->  S  e.  dom  ( nei `  J
) )
21adantl 453 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  e.  dom  ( nei `  J ) )
3 neifval.1 . . . . . . 7  |-  X  = 
U. J
43neif 17080 . . . . . 6  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )
5 fndm 5477 . . . . . 6  |-  ( ( nei `  J )  Fn  ~P X  ->  dom  ( nei `  J
)  =  ~P X
)
64, 5syl 16 . . . . 5  |-  ( J  e.  Top  ->  dom  ( nei `  J )  =  ~P X )
76eleq2d 2447 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  dom  ( nei `  J )  <->  S  e.  ~P X ) )
87adantr 452 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  -> 
( S  e.  dom  ( nei `  J )  <-> 
S  e.  ~P X
) )
92, 8mpbid 202 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  e.  ~P X
)
109elpwid 3744 1  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3256   ~Pcpw 3735   U.cuni 3950   dom cdm 4811    Fn wfn 5382   ` cfv 5387   Topctop 16874   neicnei 17077
This theorem is referenced by:  neii1  17086  neii2  17088  neiss  17089  ssnei2  17096  topssnei  17104  innei  17105  neitx  17553  cvmlift2lem12  24773  neiin  26019
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-top 16879  df-nei 17078
  Copyright terms: Public domain W3C validator