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Theorem restsspw 14701
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 3772 . . . . . . 7  |-  ( x  e.  ( Jt  A )  ->  -.  ( Jt  A
)  =  (/) )
2 restfn 14694 . . . . . . . . 9  |-t  Fn  ( _V  X.  _V )
3 fndm 5666 . . . . . . . . 9  |-  (t  Fn  ( _V  X.  _V )  ->  domt  =  ( _V  X.  _V ) )
42, 3ax-mp 5 . . . . . . . 8  |-  domt  =  ( _V  X.  _V )
54ndmov 6440 . . . . . . 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 14697 . . . . . 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 3701 . . . . . 6  |-  ( y  i^i  A )  C_  A
11 sseq1 3507 . . . . . 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 2930 . . . 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 4000 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
1614, 15sylibr 212 . 2  |-  ( x  e.  ( Jt  A )  ->  x  e.  ~P A )
1716ssriv 3490 1  |-  ( Jt  A )  C_  ~P A
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   E.wrex 2792   _Vcvv 3093    i^i cin 3457    C_ wss 3458   (/)c0 3767   ~Pcpw 3993    X. cxp 4983   dom cdm 4985    Fn wfn 5569  (class class class)co 6277   ↾t crest 14690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-rest 14692
This theorem is referenced by:  1stckgenlem  19920  prdstopn  19995  trfbas2  20210  trfil1  20253  trfil2  20254  fgtr  20257  trust  20598  zdis  21187  cnambfre  30031
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