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Theorem ssrest 17163
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 448 . . . 4  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  x  e.  ( Jt  A
) )
2 ssrexv 3352 . . . . . 6  |-  ( J 
C_  K  ->  ( E. y  e.  J  x  =  ( y  i^i  A )  ->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
32ad2antlr 708 . . . . 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 3577 . . . . . . . 8  |-  ( x  e.  ( Jt  A )  ->  -.  ( Jt  A
)  =  (/) )
5 restfn 13580 . . . . . . . . . 10  |-t  Fn  ( _V  X.  _V )
6 fndm 5485 . . . . . . . . . 10  |-  (t  Fn  ( _V  X.  _V )  ->  domt  =  ( _V  X.  _V ) )
75, 6ax-mp 8 . . . . . . . . 9  |-  domt  =  ( _V  X.  _V )
87ndmov 6171 . . . . . . . 8  |-  ( -.  ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  (/) )
94, 8nsyl2 121 . . . . . . 7  |-  ( x  e.  ( Jt  A )  ->  ( J  e. 
_V  /\  A  e.  _V ) )
109adantl 453 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( J  e.  _V  /\  A  e.  _V )
)
11 elrest 13583 . . . . . 6  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
1210, 11syl 16 . . . . 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 731 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  K  e.  V )
1410simprd 450 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  A  e.  _V )
15 elrest 13583 . . . . . 6  |-  ( ( K  e.  V  /\  A  e.  _V )  ->  ( x  e.  ( Kt  A )  <->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
1613, 14, 15syl2anc 643 . . . . 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 260 . . . 4  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Jt  A )  ->  x  e.  ( Kt  A ) ) )
181, 17mpd 15 . . 3  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  x  e.  ( Kt  A
) )
1918ex 424 . 2  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( x  e.  ( Jt  A )  ->  x  e.  ( Kt  A ) ) )
2019ssrdv 3298 1  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( Jt  A )  C_  ( Kt  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2651   _Vcvv 2900    i^i cin 3263    C_ wss 3264   (/)c0 3572    X. cxp 4817   dom cdm 4819    Fn wfn 5390  (class class class)co 6021   ↾t crest 13576
This theorem is referenced by:  1stcrest  17438  kgencmp  17499  kgencmp2  17500  kgen2ss  17509  ssufl  17872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-rest 13578
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