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Theorem topssnei 17188
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 961 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  K  e.  Top )
2 simprl 733 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  J  C_  K )
3 simpl1 960 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  J  e.  Top )
4 simprr 734 . . . . . . . 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 17170 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S ) )  ->  x  C_  X )
73, 4, 6syl2anc 643 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  C_  X )
85ntropn 17113 . . . . . . 7  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( int `  J
) `  x )  e.  J )
93, 7, 8syl2anc 643 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  e.  J )
102, 9sseldd 3349 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  e.  K )
115neiss2 17165 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
123, 4, 11syl2anc 643 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  S  C_  X )
135neiint 17168 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  X  /\  x  C_  X )  ->  (
x  e.  ( ( nei `  J ) `
 S )  <->  S  C_  (
( int `  J
) `  x )
) )
143, 12, 7, 13syl3anc 1184 . . . . . 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 202 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  S  C_  ( ( int `  J
) `  x )
)
16 opnneiss 17182 . . . . 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 1184 . . . 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 17121 . . . . 5  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( int `  J
) `  x )  C_  x )
193, 7, 18syl2anc 643 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  C_  x )
20 simpl3 962 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  X  =  Y )
217, 20sseqtrd 3384 . . . 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 17180 . . . 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 1185 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  e.  ( ( nei `  K
) `  S )
)
2524expr 599 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  J  C_  K )  ->  ( x  e.  ( ( nei `  J
) `  S )  ->  x  e.  ( ( nei `  K ) `
 S ) ) )
2625ssrdv 3354 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320   U.cuni 4015   ` cfv 5454   Topctop 16958   intcnt 17081   neicnei 17161
This theorem is referenced by:  flimss1  18005
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-top 16963  df-ntr 17084  df-nei 17162
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