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Mirrors > Home > HSE Home > Th. List > hsupval | Structured version Visualization version GIF version |
Description: Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 31291. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hsupval | ⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hilex 30881 | . . . 4 ⊢ ℋ ∈ V | |
2 | 1 | pwex 5380 | . . 3 ⊢ 𝒫 ℋ ∈ V |
3 | 2 | elpw2 5348 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ) |
4 | unieq 4920 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
5 | 4 | fveq2d 6900 | . . . 4 ⊢ (𝑥 = 𝐴 → (⊥‘∪ 𝑥) = (⊥‘∪ 𝐴)) |
6 | 5 | fveq2d 6900 | . . 3 ⊢ (𝑥 = 𝐴 → (⊥‘(⊥‘∪ 𝑥)) = (⊥‘(⊥‘∪ 𝐴))) |
7 | df-chsup 31193 | . . 3 ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥))) | |
8 | fvex 6909 | . . 3 ⊢ (⊥‘(⊥‘∪ 𝐴)) ∈ V | |
9 | 6, 7, 8 | fvmpt 7004 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) |
10 | 3, 9 | sylbir 234 | 1 ⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 𝒫 cpw 4604 ∪ cuni 4909 ‘cfv 6549 ℋchba 30801 ⊥cort 30812 ∨ℋ chsup 30816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-hilex 30881 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-chsup 31193 |
This theorem is referenced by: chsupval 31217 hsupcl 31221 hsupss 31223 hsupunss 31225 sshjval3 31236 hsupval2 31291 |
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