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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 29192. (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 28782 | . . . 4 ⊢ ℋ ∈ V | |
2 | 1 | pwex 5246 | . . 3 ⊢ 𝒫 ℋ ∈ V |
3 | 2 | elpw2 5212 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ) |
4 | unieq 4811 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
5 | 4 | fveq2d 6649 | . . . 4 ⊢ (𝑥 = 𝐴 → (⊥‘∪ 𝑥) = (⊥‘∪ 𝐴)) |
6 | 5 | fveq2d 6649 | . . 3 ⊢ (𝑥 = 𝐴 → (⊥‘(⊥‘∪ 𝑥)) = (⊥‘(⊥‘∪ 𝐴))) |
7 | df-chsup 29094 | . . 3 ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥))) | |
8 | fvex 6658 | . . 3 ⊢ (⊥‘(⊥‘∪ 𝐴)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6745 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) |
10 | 3, 9 | sylbir 238 | 1 ⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 ‘cfv 6324 ℋchba 28702 ⊥cort 28713 ∨ℋ chsup 28717 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-hilex 28782 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-chsup 29094 |
This theorem is referenced by: chsupval 29118 hsupcl 29122 hsupss 29124 hsupunss 29126 sshjval3 29137 hsupval2 29192 |
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