Theorem hsupval 28193

Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 28268. (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.

Step	Hyp	Ref	Expression
1		ax-hilex 27856	. . . 4 |- ~H e. _V
2	1	pwex 4848	. . 3 |- ~P ~H e. _V
3	2	elpw2 4828	. 2 |- ( A e. ~P ~P ~H <-> A C_ ~P ~H )
4		unieq 4444	. . . . 5 |- ( x = A -> U. x = U. A )
5	4	fveq2d 6195	. . . 4 |- ( x = A -> ( _|_ ` U. x ) = ( _|_ ` U. A ) )
6	5	fveq2d 6195	. . 3 |- ( x = A -> ( _|_ ` ( _|_ ` U. x ) ) = ( _|_ ` ( _|_ ` U. A ) ) )
7		df-chsup 28170	. . 3 |- \/H = ( x e. ~P ~P ~H |-> ( _|_ ` ( _|_ ` U. x ) ) )
8		fvex 6201	. . 3 |- ( _|_ ` ( _|_ ` U. A ) ) e. _V
9	6, 7, 8	fvmpt 6282	. 2 |- ( A e. ~P ~P ~H -> ( \/H ` A ) = ( _|_ ` ( _|_ ` U. A ) ) )
10	3, 9	sylbir 225	1 |- ( A C_ ~P ~H -> ( \/H ` A ) = ( _|_ ` ( _|_ ` U. A ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ` cfv 5888   ~Hchil 27776   _|_cort 27787   \/H chsup 27791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-chsup 28170

This theorem is referenced by:  chsupval  28194  hsupcl  28198  hsupss  28200  hsupunss  28202  sshjval3  28213  hsupval2  28268