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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 27652. (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 27240 | . . . 4 ⊢ ℋ ∈ V | |
2 | 1 | pwex 4774 | . . 3 ⊢ 𝒫 ℋ ∈ V |
3 | 2 | elpw2 4755 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ) |
4 | unieq 4380 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
5 | 4 | fveq2d 6107 | . . . 4 ⊢ (𝑥 = 𝐴 → (⊥‘∪ 𝑥) = (⊥‘∪ 𝐴)) |
6 | 5 | fveq2d 6107 | . . 3 ⊢ (𝑥 = 𝐴 → (⊥‘(⊥‘∪ 𝑥)) = (⊥‘(⊥‘∪ 𝐴))) |
7 | df-chsup 27554 | . . 3 ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥))) | |
8 | fvex 6113 | . . 3 ⊢ (⊥‘(⊥‘∪ 𝐴)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6191 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) |
10 | 3, 9 | sylbir 224 | 1 ⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 ‘cfv 5804 ℋchil 27160 ⊥cort 27171 ∨ℋ chsup 27175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-hilex 27240 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-chsup 27554 |
This theorem is referenced by: chsupval 27578 hsupcl 27582 hsupss 27584 hsupunss 27586 sshjval3 27597 hsupval2 27652 |
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