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Theorem neiint 17160
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 17159 . . . 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 17114 . . . . . . 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 3350 . . . 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 2823 . . 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 17105 . . . . . 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 17113 . . . . . 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 3362 . . . . . . 7  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( S  C_  v  <->  S  C_  ( ( int `  J ) `
 N ) ) )
19 sseq1 3361 . . . . . . 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 3044 . . . . 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 1652    e. wcel 1725   E.wrex 2698    C_ wss 3312   U.cuni 4007   ` cfv 5446   Topctop 16950   intcnt 17073   neicnei 17153
This theorem is referenced by:  opnnei  17176  topssnei  17180  iscnp4  17319  llycmpkgen2  17574  flimopn  17999  fclsneii  18041  fcfnei  18059  limcflf  19760  neiin  26326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-top 16955  df-ntr 17076  df-nei 17154
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