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Theorem ssrest 16923
Description: If  K is a finer topology than  J, then the subspace topologies induced by  A maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
ssrest  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( Jt  A )  C_  ( Kt  A ) )

Proof of Theorem ssrest
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . 4  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  x  e.  ( Jt  A
) )
2 ssrexv 3251 . . . . . 6  |-  ( J 
C_  K  ->  ( E. y  e.  J  x  =  ( y  i^i  A )  ->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
32ad2antlr 707 . . . . 5  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( E. y  e.  J  x  =  ( y  i^i  A )  ->  E. y  e.  K  x  =  ( y  i^i  A ) ) )
4 n0i 3473 . . . . . . . 8  |-  ( x  e.  ( Jt  A )  ->  -.  ( Jt  A
)  =  (/) )
5 restfn 13345 . . . . . . . . . 10  |-t  Fn  ( _V  X.  _V )
6 fndm 5359 . . . . . . . . . 10  |-  (t  Fn  ( _V  X.  _V )  ->  domt  =  ( _V  X.  _V ) )
75, 6ax-mp 8 . . . . . . . . 9  |-  domt  =  ( _V  X.  _V )
87ndmov 6020 . . . . . . . 8  |-  ( -.  ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  (/) )
94, 8nsyl2 119 . . . . . . 7  |-  ( x  e.  ( Jt  A )  ->  ( J  e. 
_V  /\  A  e.  _V ) )
109adantl 452 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( J  e.  _V  /\  A  e.  _V )
)
11 elrest 13348 . . . . . 6  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
1210, 11syl 15 . . . . 5  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
13 simpll 730 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  K  e.  V )
1410simprd 449 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  A  e.  _V )
15 elrest 13348 . . . . . 6  |-  ( ( K  e.  V  /\  A  e.  _V )  ->  ( x  e.  ( Kt  A )  <->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
1613, 14, 15syl2anc 642 . . . . 5  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Kt  A )  <->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
173, 12, 163imtr4d 259 . . . 4  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Jt  A )  ->  x  e.  ( Kt  A ) ) )
181, 17mpd 14 . . 3  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  x  e.  ( Kt  A
) )
1918ex 423 . 2  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( x  e.  ( Jt  A )  ->  x  e.  ( Kt  A ) ) )
2019ssrdv 3198 1  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( Jt  A )  C_  ( Kt  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468    X. cxp 4703   dom cdm 4705    Fn wfn 5266  (class class class)co 5874   ↾t crest 13341
This theorem is referenced by:  1stcrest  17195  kgencmp  17256  kgencmp2  17257  kgen2ss  17266  ssufl  17629
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-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-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-rest 13343
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