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Theorem restsspw 14676
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 3783 . . . . . . 7  |-  ( x  e.  ( Jt  A )  ->  -.  ( Jt  A
)  =  (/) )
2 restfn 14669 . . . . . . . . 9  |-t  Fn  ( _V  X.  _V )
3 fndm 5671 . . . . . . . . 9  |-  (t  Fn  ( _V  X.  _V )  ->  domt  =  ( _V  X.  _V ) )
42, 3ax-mp 5 . . . . . . . 8  |-  domt  =  ( _V  X.  _V )
54ndmov 6434 . . . . . . 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 14672 . . . . . 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 3712 . . . . . 6  |-  ( y  i^i  A )  C_  A
11 sseq1 3518 . . . . . 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 2945 . . . 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 4010 . . 3  |-  ( x  e.  ~P A  <->  x  C_  A
)
1614, 15sylibr 212 . 2  |-  ( x  e.  ( Jt  A )  ->  x  e.  ~P A )
1716ssriv 3501 1  |-  ( Jt  A )  C_  ~P A
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   E.wrex 2808   _Vcvv 3106    i^i cin 3468    C_ wss 3469   (/)c0 3778   ~Pcpw 4003    X. cxp 4990   dom cdm 4992    Fn wfn 5574  (class class class)co 6275   ↾t crest 14665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-rest 14667
This theorem is referenced by:  1stckgenlem  19782  prdstopn  19857  trfbas2  20072  trfil1  20115  trfil2  20116  fgtr  20119  trust  20460  zdis  21049  cnambfre  29491
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