Theorem hsupval 31422

Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 31497. (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 31087	. . . 4 ⊢ ℋ ∈ V
2	1	pwex 5327	. . 3 ⊢ 𝒫 ℋ ∈ V
3	2	elpw2 5281	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ)
4		unieq 4876	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
5	4	fveq2d 6846	. . . 4 ⊢ (𝑥 = 𝐴 → (⊥‘∪ 𝑥) = (⊥‘∪ 𝐴))
6	5	fveq2d 6846	. . 3 ⊢ (𝑥 = 𝐴 → (⊥‘(⊥‘∪ 𝑥)) = (⊥‘(⊥‘∪ 𝐴)))
7		df-chsup 31399	. . 3 ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥)))
8		fvex 6855	. . 3 ⊢ (⊥‘(⊥‘∪ 𝐴)) ∈ V
9	6, 7, 8	fvmpt 6949	. 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 1542   ∈ wcel 2114   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865  ‘cfv 6500   ℋchba 31007  ⊥cort 31018   ∨_ℋ chsup 31022

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-hilex 31087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-chsup 31399

This theorem is referenced by:  chsupval  31423  hsupcl  31427  hsupss  31429  hsupunss  31431  sshjval3  31442  hsupval2  31497