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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 29113. (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 28703 | . . . 4 ⊢ ℋ ∈ V | |
2 | 1 | pwex 5272 | . . 3 ⊢ 𝒫 ℋ ∈ V |
3 | 2 | elpw2 5239 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ) |
4 | unieq 4838 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
5 | 4 | fveq2d 6667 | . . . 4 ⊢ (𝑥 = 𝐴 → (⊥‘∪ 𝑥) = (⊥‘∪ 𝐴)) |
6 | 5 | fveq2d 6667 | . . 3 ⊢ (𝑥 = 𝐴 → (⊥‘(⊥‘∪ 𝑥)) = (⊥‘(⊥‘∪ 𝐴))) |
7 | df-chsup 29015 | . . 3 ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥))) | |
8 | fvex 6676 | . . 3 ⊢ (⊥‘(⊥‘∪ 𝐴)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6761 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) |
10 | 3, 9 | sylbir 236 | 1 ⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 ‘cfv 6348 ℋchba 28623 ⊥cort 28634 ∨ℋ chsup 28638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-hilex 28703 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-chsup 29015 |
This theorem is referenced by: chsupval 29039 hsupcl 29043 hsupss 29045 hsupunss 29047 sshjval3 29058 hsupval2 29113 |
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