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Theorem restsspw 13352
Description: The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
restsspw  |-  ( Jt  A )  C_  ~P A

Proof of Theorem restsspw
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 3473 . . . . . . 7  |-  ( x  e.  ( Jt  A )  ->  -.  ( Jt  A
)  =  (/) )
2 restfn 13345 . . . . . . . . 9  |-t  Fn  ( _V  X.  _V )
3 fndm 5359 . . . . . . . . 9  |-  (t  Fn  ( _V  X.  _V )  ->  domt  =  ( _V  X.  _V ) )
42, 3ax-mp 8 . . . . . . . 8  |-  domt  =  ( _V  X.  _V )
54ndmov 6020 . . . . . . 7  |-  ( -.  ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  (/) )
61, 5nsyl2 119 . . . . . 6  |-  ( x  e.  ( Jt  A )  ->  ( J  e. 
_V  /\  A  e.  _V ) )
7 elrest 13348 . . . . . 6  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
86, 7syl 15 . . . . 5  |-  ( x  e.  ( Jt  A )  ->  ( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
98ibi 232 . . . 4  |-  ( x  e.  ( Jt  A )  ->  E. y  e.  J  x  =  ( y  i^i  A ) )
10 inss2 3403 . . . . . 6  |-  ( y  i^i  A )  C_  A
11 sseq1 3212 . . . . . 6  |-  ( x  =  ( y  i^i 
A )  ->  (
x  C_  A  <->  ( y  i^i  A )  C_  A
) )
1210, 11mpbiri 224 . . . . 5  |-  ( x  =  ( y  i^i 
A )  ->  x  C_  A )
1312rexlimivw 2676 . . . 4  |-  ( E. y  e.  J  x  =  ( y  i^i 
A )  ->  x  C_  A )
149, 13syl 15 . . 3  |-  ( x  e.  ( Jt  A )  ->  x  C_  A
)
15 vex 2804 . . . 4  |-  x  e. 
_V
1615elpw 3644 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
1714, 16sylibr 203 . 2  |-  ( x  e.  ( Jt  A )  ->  x  e.  ~P A )
1817ssriv 3197 1  |-  ( Jt  A )  C_  ~P A
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638    X. cxp 4703   dom cdm 4705    Fn wfn 5266  (class class class)co 5874   ↾t crest 13341
This theorem is referenced by:  1stckgenlem  17264  prdstopn  17338  trfbas2  17554  trfil1  17597  trfil2  17598  fgtr  17601  zdis  18338
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-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-rest 13343
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