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Theorem topssnei 12345
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 985 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  K  e.  Top )
2 simprl 520 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  J  C_  K )
3 simpl1 984 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  J  e.  Top )
4 simprr 521 . . . . . . . 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 12330 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S ) )  ->  x  C_  X )
73, 4, 6syl2anc 408 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  C_  X )
85ntropn 12300 . . . . . . 7  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( int `  J
) `  x )  e.  J )
93, 7, 8syl2anc 408 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  e.  J )
102, 9sseldd 3098 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  e.  K )
115neiss2 12325 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
123, 4, 11syl2anc 408 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  S  C_  X )
135neiint 12328 . . . . . . 7  |-  ( ( J  e.  Top  /\  S  C_  X  /\  x  C_  X )  ->  (
x  e.  ( ( nei `  J ) `
 S )  <->  S  C_  (
( int `  J
) `  x )
) )
143, 12, 7, 13syl3anc 1216 . . . . . 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 146 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  S  C_  ( ( int `  J
) `  x )
)
16 opnneiss 12341 . . . . 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 1216 . . . 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 12304 . . . . 5  |-  ( ( J  e.  Top  /\  x  C_  X )  -> 
( ( int `  J
) `  x )  C_  x )
193, 7, 18syl2anc 408 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  (
( int `  J
) `  x )  C_  x )
20 simpl3 986 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  X  =  Y )
217, 20sseqtrd 3135 . . . 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 12340 . . . 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 1217 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  ( J  C_  K  /\  x  e.  (
( nei `  J
) `  S )
) )  ->  x  e.  ( ( nei `  K
) `  S )
)
2524expr 372 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  X  =  Y )  /\  J  C_  K )  ->  ( x  e.  ( ( nei `  J
) `  S )  ->  x  e.  ( ( nei `  K ) `
 S ) ) )
2625ssrdv 3103 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 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480    C_ wss 3071   U.cuni 3736   ` cfv 5123   Topctop 12178   intcnt 12276   neicnei 12321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-top 12179  df-ntr 12279  df-nei 12322
This theorem is referenced by: (None)
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