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Theorem neiint 14105
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 14104 . . . 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 1019 . . 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 1167 . 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 539 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  ( v  e.  J  /\  ( S  C_  v  /\  v  C_  N ) ) )  ->  S  C_  v )
61ssntr 14082 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  N  C_  X )  /\  ( v  e.  J  /\  v  C_  N ) )  ->  v  C_  ( ( int `  J
) `  N )
)
763adantl2 1156 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  ( v  e.  J  /\  v  C_  N ) )  ->  v  C_  ( ( int `  J
) `  N )
)
87adantrrl 486 . . . . 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 3180 . . . 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 2608 . . 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 1002 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  J  e.  Top )
12 simpl3 1004 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  N  C_  X )
131ntropn 14077 . . . . . 6  |-  ( ( J  e.  Top  /\  N  C_  X )  -> 
( ( int `  J
) `  N )  e.  J )
1411, 12, 13syl2anc 411 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  (
( int `  J
) `  N )  e.  J )
15 simpr 110 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  S  C_  ( ( int `  J
) `  N )
)
161ntrss2 14081 . . . . . 6  |-  ( ( J  e.  Top  /\  N  C_  X )  -> 
( ( int `  J
) `  N )  C_  N )
1711, 12, 16syl2anc 411 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  N  C_  X )  /\  S  C_  ( ( int `  J ) `  N
) )  ->  (
( int `  J
) `  N )  C_  N )
18 sseq2 3194 . . . . . . 7  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( S  C_  v  <->  S  C_  ( ( int `  J ) `
 N ) ) )
19 sseq1 3193 . . . . . . 7  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( v  C_  N  <->  ( ( int `  J ) `  N
)  C_  N )
)
2018, 19anbi12d 473 . . . . . 6  |-  ( v  =  ( ( int `  J ) `  N
)  ->  ( ( S  C_  v  /\  v  C_  N )  <->  ( S  C_  ( ( int `  J
) `  N )  /\  ( ( int `  J
) `  N )  C_  N ) ) )
2120rspcev 2856 . . . . 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 1247 . . . 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 115 . . 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 129 . 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 188 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 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   E.wrex 2469    C_ wss 3144   U.cuni 3824   ` cfv 5235   Topctop 13957   intcnt 14053   neicnei 14098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-top 13958  df-ntr 14056  df-nei 14099
This theorem is referenced by:  topssnei  14122  iscnp4  14178
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