Theorem hsupval 29804

Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 29879. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
hsupval	⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴)))

Proof of Theorem hsupval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 29469	. . . 4 ⊢ ℋ ∈ V
2	1	pwex 5316	. . 3 ⊢ 𝒫 ℋ ∈ V
3	2	elpw2 5282	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ)
4		unieq 4859	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
5	4	fveq2d 6813	. . . 4 ⊢ (𝑥 = 𝐴 → (⊥‘∪ 𝑥) = (⊥‘∪ 𝐴))
6	5	fveq2d 6813	. . 3 ⊢ (𝑥 = 𝐴 → (⊥‘(⊥‘∪ 𝑥)) = (⊥‘(⊥‘∪ 𝐴)))
7		df-chsup 29781	. . 3 ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
8		fvex 6822	. . 3 ⊢ (⊥‘(⊥‘∪ 𝐴)) ∈ V
9	6, 7, 8	fvmpt 6912	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴)))
10	3, 9	sylbir 234	1 ⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105   ⊆ wss 3896  𝒫 cpw 4543  ∪ cuni 4848  ‘cfv 6463   ℋchba 29389  ⊥cort 29400   ∨_ℋ chsup 29404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-hilex 29469

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-iota 6415  df-fun 6465  df-fv 6471  df-chsup 29781

This theorem is referenced by:  chsupval  29805  hsupcl  29809  hsupss  29811  hsupunss  29813  sshjval3  29824  hsupval2  29879