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Theorem hsupval 31216
Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 31291. (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 30881 . . . 4 ℋ ∈ V
21pwex 5380 . . 3 𝒫 ℋ ∈ V
32elpw2 5348 . 2 (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ)
4 unieq 4920 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
54fveq2d 6900 . . . 4 (𝑥 = 𝐴 → (⊥‘ 𝑥) = (⊥‘ 𝐴))
65fveq2d 6900 . . 3 (𝑥 = 𝐴 → (⊥‘(⊥‘ 𝑥)) = (⊥‘(⊥‘ 𝐴)))
7 df-chsup 31193 . . 3 = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘ 𝑥)))
8 fvex 6909 . . 3 (⊥‘(⊥‘ 𝐴)) ∈ V
96, 7, 8fvmpt 7004 . 2 (𝐴 ∈ 𝒫 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))
103, 9sylbir 234 1 (𝐴 ⊆ 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wss 3944  𝒫 cpw 4604   cuni 4909  cfv 6549  chba 30801  cort 30812   chsup 30816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-hilex 30881
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-chsup 31193
This theorem is referenced by:  chsupval  31217  hsupcl  31221  hsupss  31223  hsupunss  31225  sshjval3  31236  hsupval2  31291
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