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Theorem neiint 12151
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 12150 . . . 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 982 . . 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 1130 . 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 511 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  ( v  e.  J  /\  ( S  C_  v  /\  v  C_  N ) ) )  ->  S  C_  v )
61ssntr 12128 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  N  C_  X )  /\  ( v  e.  J  /\  v  C_  N ) )  ->  v  C_  ( ( int `  J
) `  N )
)
763adantl2 1119 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  ( v  e.  J  /\  v  C_  N ) )  ->  v  C_  ( ( int `  J
) `  N )
)
87adantrrl 475 . . . . 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 3071 . . . 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 2522 . . 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 965 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  J  e.  Top )
12 simpl3 967 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  N  C_  X )
131ntropn 12123 . . . . . 6  |-  ( ( J  e.  Top  /\  N  C_  X )  -> 
( ( int `  J
) `  N )  e.  J )
1411, 12, 13syl2anc 406 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  (
( int `  J
) `  N )  e.  J )
15 simpr 109 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  S  C_  ( ( int `  J
) `  N )
)
161ntrss2 12127 . . . . . 6  |-  ( ( J  e.  Top  /\  N  C_  X )  -> 
( ( int `  J
) `  N )  C_  N )
1711, 12, 16syl2anc 406 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  (
( int `  J
) `  N )  C_  N )
18 sseq2 3085 . . . . . . 7  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( S  C_  v  <->  S  C_  ( ( int `  J ) `
 N ) ) )
19 sseq1 3084 . . . . . . 7  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( v  C_  N  <->  ( ( int `  J ) `  N
)  C_  N )
)
2018, 19anbi12d 462 . . . . . 6  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( ( S  C_  v  /\  v  C_  N )  <->  ( S  C_  ( ( int `  J
) `  N )  /\  ( ( int `  J
) `  N )  C_  N ) ) )
2120rspcev 2758 . . . . 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 1195 . . . 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 114 . . 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 128 . 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 187 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    /\ wa 103    <-> wb 104    /\ w3a 943    = wceq 1312    e. wcel 1461   E.wrex 2389    C_ wss 3035   U.cuni 3700   ` cfv 5079   Topctop 12001   intcnt 12099   neicnei 12144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-top 12002  df-ntr 12102  df-nei 12145
This theorem is referenced by:  topssnei  12168  iscnp4  12223
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