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Theorem topssnei 16877
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 959 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  K  e.  Top )
2 simprl 732 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  J  C_  K )
3 simpl1 958 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  J  e.  Top )
4 simprr 733 . . . . . . . 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 16859 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S ) )  ->  x  C_  X )
73, 4, 6syl2anc 642 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  C_  X )
85ntropn 16802 . . . . . . 7  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( int `  J
) `  x )  e.  J )
93, 7, 8syl2anc 642 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  e.  J )
102, 9sseldd 3194 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  e.  K )
115neiss2 16854 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
123, 4, 11syl2anc 642 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  S  C_  X )
135neiint 16857 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  X  /\  x  C_  X )  ->  (
x  e.  ( ( nei `  J ) `
 S )  <->  S  C_  (
( int `  J
) `  x )
) )
143, 12, 7, 13syl3anc 1182 . . . . . 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 201 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  S  C_  ( ( int `  J
) `  x )
)
16 opnneiss 16871 . . . . 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 1182 . . . 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 16810 . . . . 5  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( int `  J
) `  x )  C_  x )
193, 7, 18syl2anc 642 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  C_  x )
20 simpl3 960 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  X  =  Y )
217, 20sseqtrd 3227 . . . 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 16869 . . . 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 1183 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  e.  ( ( nei `  K
) `  S )
)
2524expr 598 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  J  C_  K )  ->  ( x  e.  ( ( nei `  J
) `  S )  ->  x  e.  ( ( nei `  K ) `
 S ) ) )
2625ssrdv 3198 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   U.cuni 3843   ` cfv 5271   Topctop 16647   intcnt 16770   neicnei 16850
This theorem is referenced by:  flimss1  17684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-ntr 16773  df-nei 16851
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