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Theorem hsupval 29038
Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 29113. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hsupval (𝐴 ⊆ 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))

Proof of Theorem hsupval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 28703 . . . 4 ℋ ∈ V
21pwex 5272 . . 3 𝒫 ℋ ∈ V
32elpw2 5239 . 2 (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ)
4 unieq 4838 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
54fveq2d 6667 . . . 4 (𝑥 = 𝐴 → (⊥‘ 𝑥) = (⊥‘ 𝐴))
65fveq2d 6667 . . 3 (𝑥 = 𝐴 → (⊥‘(⊥‘ 𝑥)) = (⊥‘(⊥‘ 𝐴)))
7 df-chsup 29015 . . 3 = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘ 𝑥)))
8 fvex 6676 . . 3 (⊥‘(⊥‘ 𝐴)) ∈ V
96, 7, 8fvmpt 6761 . 2 (𝐴 ∈ 𝒫 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))
103, 9sylbir 236 1 (𝐴 ⊆ 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wss 3933  𝒫 cpw 4535   cuni 4830  cfv 6348  chba 28623  cort 28634   chsup 28638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-chsup 29015
This theorem is referenced by:  chsupval  29039  hsupcl  29043  hsupss  29045  hsupunss  29047  sshjval3  29058  hsupval2  29113
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