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Theorem hsupval 28779
 Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 28854. (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 28442 . . . 4 ℋ ∈ V
21pwex 5092 . . 3 𝒫 ℋ ∈ V
32elpw2 5062 . 2 (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ)
4 unieq 4679 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
54fveq2d 6450 . . . 4 (𝑥 = 𝐴 → (⊥‘ 𝑥) = (⊥‘ 𝐴))
65fveq2d 6450 . . 3 (𝑥 = 𝐴 → (⊥‘(⊥‘ 𝑥)) = (⊥‘(⊥‘ 𝐴)))
7 df-chsup 28756 . . 3 = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘ 𝑥)))
8 fvex 6459 . . 3 (⊥‘(⊥‘ 𝐴)) ∈ V
96, 7, 8fvmpt 6542 . 2 (𝐴 ∈ 𝒫 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))
103, 9sylbir 227 1 (𝐴 ⊆ 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2106   ⊆ wss 3791  𝒫 cpw 4378  ∪ cuni 4671  ‘cfv 6135   ℋchba 28362  ⊥cort 28373   ∨ℋ chsup 28377 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-hilex 28442 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-chsup 28756 This theorem is referenced by:  chsupval  28780  hsupcl  28784  hsupss  28786  hsupunss  28788  sshjval3  28799  hsupval2  28854
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