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Theorem hsupval 28321
Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 28396. (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 27984 . . . 4 ℋ ∈ V
21pwex 4878 . . 3 𝒫 ℋ ∈ V
32elpw2 4858 . 2 (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ)
4 unieq 4476 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
54fveq2d 6233 . . . 4 (𝑥 = 𝐴 → (⊥‘ 𝑥) = (⊥‘ 𝐴))
65fveq2d 6233 . . 3 (𝑥 = 𝐴 → (⊥‘(⊥‘ 𝑥)) = (⊥‘(⊥‘ 𝐴)))
7 df-chsup 28298 . . 3 = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘ 𝑥)))
8 fvex 6239 . . 3 (⊥‘(⊥‘ 𝐴)) ∈ V
96, 7, 8fvmpt 6321 . 2 (𝐴 ∈ 𝒫 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))
103, 9sylbir 225 1 (𝐴 ⊆ 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wss 3607  𝒫 cpw 4191   cuni 4468  cfv 5926  chil 27904  cort 27915   chsup 27919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-hilex 27984
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-chsup 28298
This theorem is referenced by:  chsupval  28322  hsupcl  28326  hsupss  28328  hsupunss  28330  sshjval3  28341  hsupval2  28396
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