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Theorem hsupval 25956
Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 26031. (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 25620 . . . 4  |-  ~H  e.  _V
21pwex 4630 . . 3  |-  ~P ~H  e.  _V
32elpw2 4611 . 2  |-  ( A  e.  ~P ~P ~H  <->  A 
C_  ~P ~H )
4 unieq 4253 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
54fveq2d 5870 . . . 4  |-  ( x  =  A  ->  ( _|_ `  U. x )  =  ( _|_ `  U. A ) )
65fveq2d 5870 . . 3  |-  ( x  =  A  ->  ( _|_ `  ( _|_ `  U. x ) )  =  ( _|_ `  ( _|_ `  U. A ) ) )
7 df-chsup 25933 . . 3  |-  \/H  =  ( x  e.  ~P ~P ~H  |->  ( _|_ `  ( _|_ `  U. x ) ) )
8 fvex 5876 . . 3  |-  ( _|_ `  ( _|_ `  U. A ) )  e. 
_V
96, 7, 8fvmpt 5950 . 2  |-  ( A  e.  ~P ~P ~H  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )
103, 9sylbir 213 1  |-  ( A 
C_  ~P ~H  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   ` cfv 5588   ~Hchil 25540   _|_cort 25551    \/H chsup 25555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-hilex 25620
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-chsup 25933
This theorem is referenced by:  chsupval  25957  hsupcl  25961  hsupss  25963  hsupunss  25965  sshjval3  25976  hsupval2  26031
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