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Theorem restsspw 14474
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 3742 . . . . . . 7  |-  ( x  e.  ( Jt  A )  ->  -.  ( Jt  A
)  =  (/) )
2 restfn 14467 . . . . . . . . 9  |-t  Fn  ( _V  X.  _V )
3 fndm 5610 . . . . . . . . 9  |-  (t  Fn  ( _V  X.  _V )  ->  domt  =  ( _V  X.  _V ) )
42, 3ax-mp 5 . . . . . . . 8  |-  domt  =  ( _V  X.  _V )
54ndmov 6349 . . . . . . 7  |-  ( -.  ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  (/) )
61, 5nsyl2 127 . . . . . 6  |-  ( x  e.  ( Jt  A )  ->  ( J  e. 
_V  /\  A  e.  _V ) )
7 elrest 14470 . . . . . 6  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
86, 7syl 16 . . . . 5  |-  ( x  e.  ( Jt  A )  ->  ( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
98ibi 241 . . . 4  |-  ( x  e.  ( Jt  A )  ->  E. y  e.  J  x  =  ( y  i^i  A ) )
10 inss2 3671 . . . . . 6  |-  ( y  i^i  A )  C_  A
11 sseq1 3477 . . . . . 6  |-  ( x  =  ( y  i^i 
A )  ->  (
x  C_  A  <->  ( y  i^i  A )  C_  A
) )
1210, 11mpbiri 233 . . . . 5  |-  ( x  =  ( y  i^i 
A )  ->  x  C_  A )
1312rexlimivw 2935 . . . 4  |-  ( E. y  e.  J  x  =  ( y  i^i 
A )  ->  x  C_  A )
149, 13syl 16 . . 3  |-  ( x  e.  ( Jt  A )  ->  x  C_  A
)
15 selpw 3967 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
1614, 15sylibr 212 . 2  |-  ( x  e.  ( Jt  A )  ->  x  e.  ~P A )
1716ssriv 3460 1  |-  ( Jt  A )  C_  ~P A
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2796   _Vcvv 3070    i^i cin 3427    C_ wss 3428   (/)c0 3737   ~Pcpw 3960    X. cxp 4938   dom cdm 4940    Fn wfn 5513  (class class class)co 6192   ↾t crest 14463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-rest 14465
This theorem is referenced by:  1stckgenlem  19244  prdstopn  19319  trfbas2  19534  trfil1  19577  trfil2  19578  fgtr  19581  trust  19922  zdis  20511  cnambfre  28580
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