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Theorem cssval 16940
Description: The set of closed subspaces of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
cssval.o  |-  ._|_  =  ( ocv `  W )
cssval.c  |-  C  =  ( CSubSp `  W )
Assertion
Ref Expression
cssval  |-  ( W  e.  X  ->  C  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
Distinct variable groups:    ._|_ , s    W, s
Allowed substitution hints:    C( s)    X( s)

Proof of Theorem cssval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elex 2970 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 cssval.c . . 3  |-  C  =  ( CSubSp `  W )
3 fveq2 5757 . . . . . . . 8  |-  ( w  =  W  ->  ( ocv `  w )  =  ( ocv `  W
) )
4 cssval.o . . . . . . . 8  |-  ._|_  =  ( ocv `  W )
53, 4syl6eqr 2492 . . . . . . 7  |-  ( w  =  W  ->  ( ocv `  w )  = 
._|_  )
65fveq1d 5759 . . . . . . 7  |-  ( w  =  W  ->  (
( ocv `  w
) `  s )  =  (  ._|_  `  s
) )
75, 6fveq12d 5763 . . . . . 6  |-  ( w  =  W  ->  (
( ocv `  w
) `  ( ( ocv `  w ) `  s ) )  =  (  ._|_  `  (  ._|_  `  s ) ) )
87eqeq2d 2453 . . . . 5  |-  ( w  =  W  ->  (
s  =  ( ( ocv `  w ) `
 ( ( ocv `  w ) `  s
) )  <->  s  =  (  ._|_  `  (  ._|_  `  s ) ) ) )
98abbidv 2556 . . . 4  |-  ( w  =  W  ->  { s  |  s  =  ( ( ocv `  w
) `  ( ( ocv `  w ) `  s ) ) }  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
10 df-css 16922 . . . 4  |-  CSubSp  =  ( w  e.  _V  |->  { s  |  s  =  ( ( ocv `  w
) `  ( ( ocv `  w ) `  s ) ) } )
11 fvex 5771 . . . . . 6  |-  ( Base `  W )  e.  _V
1211pwex 4411 . . . . 5  |-  ~P ( Base `  W )  e. 
_V
13 id 21 . . . . . . 7  |-  ( s  =  (  ._|_  `  (  ._|_  `  s ) )  ->  s  =  ( 
._|_  `  (  ._|_  `  s
) ) )
14 eqid 2442 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
1514, 4ocvss 16928 . . . . . . . 8  |-  (  ._|_  `  (  ._|_  `  s ) )  C_  ( Base `  W )
16 fvex 5771 . . . . . . . . 9  |-  (  ._|_  `  (  ._|_  `  s ) )  e.  _V
1716elpw 3829 . . . . . . . 8  |-  ( ( 
._|_  `  (  ._|_  `  s
) )  e.  ~P ( Base `  W )  <->  ( 
._|_  `  (  ._|_  `  s
) )  C_  ( Base `  W ) )
1815, 17mpbir 202 . . . . . . 7  |-  (  ._|_  `  (  ._|_  `  s ) )  e.  ~P ( Base `  W )
1913, 18syl6eqel 2530 . . . . . 6  |-  ( s  =  (  ._|_  `  (  ._|_  `  s ) )  ->  s  e.  ~P ( Base `  W )
)
2019abssi 3404 . . . . 5  |-  { s  |  s  =  ( 
._|_  `  (  ._|_  `  s
) ) }  C_  ~P ( Base `  W
)
2112, 20ssexi 4377 . . . 4  |-  { s  |  s  =  ( 
._|_  `  (  ._|_  `  s
) ) }  e.  _V
229, 10, 21fvmpt 5835 . . 3  |-  ( W  e.  _V  ->  ( CSubSp `
 W )  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
232, 22syl5eq 2486 . 2  |-  ( W  e.  _V  ->  C  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
241, 23syl 16 1  |-  ( W  e.  X  ->  C  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727   {cab 2428   _Vcvv 2962    C_ wss 3306   ~Pcpw 3823   ` cfv 5483   Basecbs 13500   ocvcocv 16918   CSubSpccss 16919
This theorem is referenced by:  iscss  16941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491  df-ov 6113  df-ocv 16921  df-css 16922
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