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Theorem topssnei 14139
Description: A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)
Hypotheses
Ref Expression
tpnei.1  |-  X  = 
U. J
topssnei.2  |-  Y  = 
U. K
Assertion
Ref Expression
topssnei  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  J  C_  K )  ->  ( ( nei `  J ) `  S
)  C_  ( ( nei `  K ) `  S ) )

Proof of Theorem topssnei
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl2 1003 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  K  e.  Top )
2 simprl 529 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  J  C_  K )
3 simpl1 1002 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  J  e.  Top )
4 simprr 531 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  e.  ( ( nei `  J
) `  S )
)
5 tpnei.1 . . . . . . . . 9  |-  X  = 
U. J
65neii1 14124 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S ) )  ->  x  C_  X )
73, 4, 6syl2anc 411 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  C_  X )
85ntropn 14094 . . . . . . 7  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( int `  J
) `  x )  e.  J )
93, 7, 8syl2anc 411 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  e.  J )
102, 9sseldd 3171 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  e.  K )
115neiss2 14119 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
123, 4, 11syl2anc 411 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  S  C_  X )
135neiint 14122 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  X  /\  x  C_  X )  ->  (
x  e.  ( ( nei `  J ) `
 S )  <->  S  C_  (
( int `  J
) `  x )
) )
143, 12, 7, 13syl3anc 1249 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
x  e.  ( ( nei `  J ) `
 S )  <->  S  C_  (
( int `  J
) `  x )
) )
154, 14mpbid 147 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  S  C_  ( ( int `  J
) `  x )
)
16 opnneiss 14135 . . . . 5  |-  ( ( K  e.  Top  /\  ( ( int `  J
) `  x )  e.  K  /\  S  C_  ( ( int `  J
) `  x )
)  ->  ( ( int `  J ) `  x )  e.  ( ( nei `  K
) `  S )
)
171, 10, 15, 16syl3anc 1249 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  e.  ( ( nei `  K
) `  S )
)
185ntrss2 14098 . . . . 5  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( int `  J
) `  x )  C_  x )
193, 7, 18syl2anc 411 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  C_  x )
20 simpl3 1004 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  X  =  Y )
217, 20sseqtrd 3208 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  C_  Y )
22 topssnei.2 . . . . 5  |-  Y  = 
U. K
2322ssnei2 14134 . . . 4  |-  ( ( ( K  e.  Top  /\  ( ( int `  J
) `  x )  e.  ( ( nei `  K
) `  S )
)  /\  ( (
( int `  J
) `  x )  C_  x  /\  x  C_  Y ) )  ->  x  e.  ( ( nei `  K ) `  S ) )
241, 17, 19, 21, 23syl22anc 1250 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  e.  ( ( nei `  K
) `  S )
)
2524expr 375 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  J  C_  K )  ->  ( x  e.  ( ( nei `  J
) `  S )  ->  x  e.  ( ( nei `  K ) `
 S ) ) )
2625ssrdv 3176 1  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  J  C_  K )  ->  ( ( nei `  J ) `  S
)  C_  ( ( nei `  K ) `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160    C_ wss 3144   U.cuni 3824   ` cfv 5235   Topctop 13974   intcnt 14070   neicnei 14115
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 13975  df-ntr 14073  df-nei 14116
This theorem is referenced by: (None)
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