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Theorem hsupval 21929
Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 22004. (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 21595 . . . 4  |-  ~H  e.  _V
21pwex 4209 . . 3  |-  ~P ~H  e.  _V
32elpw2 4191 . 2  |-  ( A  e.  ~P ~P ~H  <->  A 
C_  ~P ~H )
4 unieq 3852 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
54fveq2d 5545 . . . 4  |-  ( x  =  A  ->  ( _|_ `  U. x )  =  ( _|_ `  U. A ) )
65fveq2d 5545 . . 3  |-  ( x  =  A  ->  ( _|_ `  ( _|_ `  U. x ) )  =  ( _|_ `  ( _|_ `  U. A ) ) )
7 df-chsup 21906 . . 3  |-  \/H  =  ( x  e.  ~P ~P ~H  |->  ( _|_ `  ( _|_ `  U. x ) ) )
8 fvex 5555 . . 3  |-  ( _|_ `  ( _|_ `  U. A ) )  e. 
_V
96, 7, 8fvmpt 5618 . 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 1632    e. wcel 1696    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   ` cfv 5271   ~Hchil 21515   _|_cort 21526    \/H chsup 21530
This theorem is referenced by:  chsupval  21930  hsupcl  21934  hsupss  21936  hsupunss  21938  sshjval3  21949  hsupval2  22004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-chsup 21906
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