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Theorem hsupval 29095
Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 29170. (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 28760 . . . 4 ℋ ∈ V
21pwex 5254 . . 3 𝒫 ℋ ∈ V
32elpw2 5221 . 2 (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ)
4 unieq 4822 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
54fveq2d 6647 . . . 4 (𝑥 = 𝐴 → (⊥‘ 𝑥) = (⊥‘ 𝐴))
65fveq2d 6647 . . 3 (𝑥 = 𝐴 → (⊥‘(⊥‘ 𝑥)) = (⊥‘(⊥‘ 𝐴)))
7 df-chsup 29072 . . 3 = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘ 𝑥)))
8 fvex 6656 . . 3 (⊥‘(⊥‘ 𝐴)) ∈ V
96, 7, 8fvmpt 6741 . 2 (𝐴 ∈ 𝒫 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))
103, 9sylbir 238 1 (𝐴 ⊆ 𝒫 ℋ → ( 𝐴) = (⊥‘(⊥‘ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wss 3910  𝒫 cpw 4512   cuni 4811  cfv 6328  chba 28680  cort 28691   chsup 28695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-hilex 28760
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-chsup 29072
This theorem is referenced by:  chsupval  29096  hsupcl  29100  hsupss  29102  hsupunss  29104  sshjval3  29115  hsupval2  29170
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