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Theorem hsupval 22841
Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 22916. (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 22507 . . . 4  |-  ~H  e.  _V
21pwex 4385 . . 3  |-  ~P ~H  e.  _V
32elpw2 4367 . 2  |-  ( A  e.  ~P ~P ~H  <->  A 
C_  ~P ~H )
4 unieq 4026 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
54fveq2d 5735 . . . 4  |-  ( x  =  A  ->  ( _|_ `  U. x )  =  ( _|_ `  U. A ) )
65fveq2d 5735 . . 3  |-  ( x  =  A  ->  ( _|_ `  ( _|_ `  U. x ) )  =  ( _|_ `  ( _|_ `  U. A ) ) )
7 df-chsup 22818 . . 3  |-  \/H  =  ( x  e.  ~P ~P ~H  |->  ( _|_ `  ( _|_ `  U. x ) ) )
8 fvex 5745 . . 3  |-  ( _|_ `  ( _|_ `  U. A ) )  e. 
_V
96, 7, 8fvmpt 5809 . 2  |-  ( A  e.  ~P ~P ~H  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )
103, 9sylbir 206 1  |-  ( A 
C_  ~P ~H  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   ` cfv 5457   ~Hchil 22427   _|_cort 22438    \/H chsup 22442
This theorem is referenced by:  chsupval  22842  hsupcl  22846  hsupss  22848  hsupunss  22850  sshjval3  22861  hsupval2  22916
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 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-hilex 22507
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-chsup 22818
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