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Theorem neiint 17093
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 17092 . . . 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 977 . . 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 1125 . 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 741 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  ( v  e.  J  /\  ( S  C_  v  /\  v  C_  N ) ) )  ->  S  C_  v )
61ssntr 17047 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  N  C_  X )  /\  ( v  e.  J  /\  v  C_  N ) )  ->  v  C_  ( ( int `  J
) `  N )
)
763adantl2 1114 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  ( v  e.  J  /\  v  C_  N ) )  ->  v  C_  ( ( int `  J
) `  N )
)
87adantrrl 705 . . . . 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 3303 . . . 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 2776 . . 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 960 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  J  e.  Top )
12 simpl3 962 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  N  C_  X )
131ntropn 17038 . . . . . 6  |-  ( ( J  e.  Top  /\  N  C_  X )  -> 
( ( int `  J
) `  N )  e.  J )
1411, 12, 13syl2anc 643 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  (
( int `  J
) `  N )  e.  J )
15 simpr 448 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  S  C_  ( ( int `  J
) `  N )
)
161ntrss2 17046 . . . . . 6  |-  ( ( J  e.  Top  /\  N  C_  X )  -> 
( ( int `  J
) `  N )  C_  N )
1711, 12, 16syl2anc 643 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  (
( int `  J
) `  N )  C_  N )
18 sseq2 3315 . . . . . . 7  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( S  C_  v  <->  S  C_  ( ( int `  J ) `
 N ) ) )
19 sseq1 3314 . . . . . . 7  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( v  C_  N  <->  ( ( int `  J ) `  N
)  C_  N )
)
2018, 19anbi12d 692 . . . . . 6  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( ( S  C_  v  /\  v  C_  N )  <->  ( S  C_  ( ( int `  J
) `  N )  /\  ( ( int `  J
) `  N )  C_  N ) ) )
2120rspcev 2997 . . . . 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 1182 . . . 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 424 . . 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 184 . 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 245 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  ->  ( N  e.  ( ( nei `  J ) `  S )  <->  S  C_  (
( int `  J
) `  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2652    C_ wss 3265   U.cuni 3959   ` cfv 5396   Topctop 16883   intcnt 17006   neicnei 17086
This theorem is referenced by:  opnnei  17109  topssnei  17113  iscnp4  17251  llycmpkgen2  17505  flimopn  17930  fclsneii  17972  fcfnei  17990  limcflf  19637  neiin  26028
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-top 16888  df-ntr 17009  df-nei 17087
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