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

Theorem neiint 19714
Description: An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
neifval.1  |-  X  = 
U. J
Assertion
Ref Expression
neiint  |-  ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  ->  ( N  e.  ( ( nei `  J ) `  S )  <->  S  C_  (
( int `  J
) `  N )
) )

Proof of Theorem neiint
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . . 5  |-  X  = 
U. J
21isnei 19713 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  <->  ( N  C_  X  /\  E. v  e.  J  ( S  C_  v  /\  v  C_  N ) ) ) )
323adant3 1014 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  ->  ( N  e.  ( ( nei `  J ) `  S )  <->  ( N  C_  X  /\  E. v  e.  J  ( S  C_  v  /\  v  C_  N ) ) ) )
433anibar 1162 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  ->  ( N  e.  ( ( nei `  J ) `  S )  <->  E. v  e.  J  ( S  C_  v  /\  v  C_  N ) ) )
5 simprrl 763 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  ( v  e.  J  /\  ( S  C_  v  /\  v  C_  N ) ) )  ->  S  C_  v )
61ssntr 19667 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  N  C_  X )  /\  ( v  e.  J  /\  v  C_  N ) )  ->  v  C_  ( ( int `  J
) `  N )
)
763adantl2 1151 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  ( v  e.  J  /\  v  C_  N ) )  ->  v  C_  ( ( int `  J
) `  N )
)
87adantrrl 721 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  ( v  e.  J  /\  ( S  C_  v  /\  v  C_  N ) ) )  ->  v  C_  ( ( int `  J
) `  N )
)
95, 8sstrd 3444 . . . 4  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  ( v  e.  J  /\  ( S  C_  v  /\  v  C_  N ) ) )  ->  S  C_  ( ( int `  J
) `  N )
)
109rexlimdvaa 2889 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  ->  ( E. v  e.  J  ( S  C_  v  /\  v  C_  N )  ->  S  C_  ( ( int `  J ) `  N
) ) )
11 simpl1 997 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  J  e.  Top )
12 simpl3 999 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  N  C_  X )
131ntropn 19658 . . . . . 6  |-  ( ( J  e.  Top  /\  N  C_  X )  -> 
( ( int `  J
) `  N )  e.  J )
1411, 12, 13syl2anc 659 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  (
( int `  J
) `  N )  e.  J )
15 simpr 459 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  S  C_  ( ( int `  J
) `  N )
)
161ntrss2 19666 . . . . . 6  |-  ( ( J  e.  Top  /\  N  C_  X )  -> 
( ( int `  J
) `  N )  C_  N )
1711, 12, 16syl2anc 659 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  (
( int `  J
) `  N )  C_  N )
18 sseq2 3456 . . . . . . 7  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( S  C_  v  <->  S  C_  ( ( int `  J ) `
 N ) ) )
19 sseq1 3455 . . . . . . 7  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( v  C_  N  <->  ( ( int `  J ) `  N
)  C_  N )
)
2018, 19anbi12d 708 . . . . . 6  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( ( S  C_  v  /\  v  C_  N )  <->  ( S  C_  ( ( int `  J
) `  N )  /\  ( ( int `  J
) `  N )  C_  N ) ) )
2120rspcev 3152 . . . . 5  |-  ( ( ( ( int `  J
) `  N )  e.  J  /\  ( S  C_  ( ( int `  J ) `  N
)  /\  ( ( int `  J ) `  N )  C_  N
) )  ->  E. v  e.  J  ( S  C_  v  /\  v  C_  N ) )
2214, 15, 17, 21syl12anc 1224 . . . 4  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  E. v  e.  J  ( S  C_  v  /\  v  C_  N ) )
2322ex 432 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  ->  ( S  C_  ( ( int `  J ) `  N
)  ->  E. v  e.  J  ( S  C_  v  /\  v  C_  N ) ) )
2410, 23impbid 191 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  ->  ( E. v  e.  J  ( S  C_  v  /\  v  C_  N )  <->  S  C_  (
( int `  J
) `  N )
) )
254, 24bitrd 253 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  ->  ( N  e.  ( ( nei `  J ) `  S )  <->  S  C_  (
( int `  J
) `  N )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   E.wrex 2747    C_ wss 3406   U.cuni 4180   ` cfv 5513   Topctop 19502   intcnt 19626   neicnei 19707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-top 19507  df-ntr 19629  df-nei 19708
This theorem is referenced by:  opnnei  19730  topssnei  19734  iscnp4  19873  llycmpkgen2  20159  flimopn  20584  fclsneii  20626  fcfnei  20644  limcflf  22393  neiin  30356
  Copyright terms: Public domain W3C validator