Theorem hsupval 31270

Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 31345. (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 30935	. . . 4 ⊢ ℋ ∈ V
2	1	pwex 5338	. . 3 ⊢ 𝒫 ℋ ∈ V
3	2	elpw2 5292	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ)
4		unieq 4885	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
5	4	fveq2d 6865	. . . 4 ⊢ (𝑥 = 𝐴 → (⊥‘∪ 𝑥) = (⊥‘∪ 𝐴))
6	5	fveq2d 6865	. . 3 ⊢ (𝑥 = 𝐴 → (⊥‘(⊥‘∪ 𝑥)) = (⊥‘(⊥‘∪ 𝐴)))
7		df-chsup 31247	. . 3 ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
8		fvex 6874	. . 3 ⊢ (⊥‘(⊥‘∪ 𝐴)) ∈ V
9	6, 7, 8	fvmpt 6971	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴)))
10	3, 9	sylbir 235	1 ⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874  ‘cfv 6514   ℋchba 30855  ⊥cort 30866   ∨_ℋ chsup 30870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-hilex 30935

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-chsup 31247

This theorem is referenced by:  chsupval  31271  hsupcl  31275  hsupss  31277  hsupunss  31279  sshjval3  31290  hsupval2  31345