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Theorem restsspw 14362
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 3637 . . . . . . 7  |-  ( x  e.  ( Jt  A )  ->  -.  ( Jt  A
)  =  (/) )
2 restfn 14355 . . . . . . . . 9  |-t  Fn  ( _V  X.  _V )
3 fndm 5505 . . . . . . . . 9  |-  (t  Fn  ( _V  X.  _V )  ->  domt  =  ( _V  X.  _V ) )
42, 3ax-mp 5 . . . . . . . 8  |-  domt  =  ( _V  X.  _V )
54ndmov 6242 . . . . . . 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 14358 . . . . . 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 3566 . . . . . 6  |-  ( y  i^i  A )  C_  A
11 sseq1 3372 . . . . . 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 2832 . . . 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 3862 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
1614, 15sylibr 212 . 2  |-  ( x  e.  ( Jt  A )  ->  x  e.  ~P A )
1716ssriv 3355 1  |-  ( Jt  A )  C_  ~P A
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2711   _Vcvv 2967    i^i cin 3322    C_ wss 3323   (/)c0 3632   ~Pcpw 3855    X. cxp 4833   dom cdm 4835    Fn wfn 5408  (class class class)co 6086   ↾t crest 14351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-rest 14353
This theorem is referenced by:  1stckgenlem  19106  prdstopn  19181  trfbas2  19396  trfil1  19439  trfil2  19440  fgtr  19443  trust  19784  zdis  20373  cnambfre  28411
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