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Theorem hsupval 21913
Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 21988. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hsupval  |-  ( A 
C_  ~P ~H  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )

Proof of Theorem hsupval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 21579 . . . 4  |-  ~H  e.  _V
21pwex 4193 . . 3  |-  ~P ~H  e.  _V
32elpw2 4175 . 2  |-  ( A  e.  ~P ~P ~H  <->  A 
C_  ~P ~H )
4 unieq 3836 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
54fveq2d 5529 . . . 4  |-  ( x  =  A  ->  ( _|_ `  U. x )  =  ( _|_ `  U. A ) )
65fveq2d 5529 . . 3  |-  ( x  =  A  ->  ( _|_ `  ( _|_ `  U. x ) )  =  ( _|_ `  ( _|_ `  U. A ) ) )
7 df-chsup 21890 . . 3  |-  \/H  =  ( x  e.  ~P ~P ~H  |->  ( _|_ `  ( _|_ `  U. x ) ) )
8 fvex 5539 . . 3  |-  ( _|_ `  ( _|_ `  U. A ) )  e. 
_V
96, 7, 8fvmpt 5602 . 2  |-  ( A  e.  ~P ~P ~H  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )
103, 9sylbir 204 1  |-  ( A 
C_  ~P ~H  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   ` cfv 5255   ~Hchil 21499   _|_cort 21510    \/H chsup 21514
This theorem is referenced by:  chsupval  21914  hsupcl  21918  hsupss  21920  hsupunss  21922  sshjval3  21933  hsupval2  21988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-chsup 21890
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