Theorem hsupval 30855

Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 30930. (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 30520	. . . 4 ⊢ ℋ ∈ V
2	1	pwex 5378	. . 3 ⊢ 𝒫 ℋ ∈ V
3	2	elpw2 5345	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ)
4		unieq 4919	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
5	4	fveq2d 6895	. . . 4 ⊢ (𝑥 = 𝐴 → (⊥‘∪ 𝑥) = (⊥‘∪ 𝐴))
6	5	fveq2d 6895	. . 3 ⊢ (𝑥 = 𝐴 → (⊥‘(⊥‘∪ 𝑥)) = (⊥‘(⊥‘∪ 𝐴)))
7		df-chsup 30832	. . 3 ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
8		fvex 6904	. . 3 ⊢ (⊥‘(⊥‘∪ 𝐴)) ∈ V
9	6, 7, 8	fvmpt 6998	. 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 3948  𝒫 cpw 4602  ∪ cuni 4908  ‘cfv 6543   ℋchba 30440  ⊥cort 30451   ∨_ℋ chsup 30455

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 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-hilex 30520

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-chsup 30832

This theorem is referenced by:  chsupval  30856  hsupcl  30860  hsupss  30862  hsupunss  30864  sshjval3  30875  hsupval2  30930