Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 29186. (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 28776 | . . . 4 ⊢ ℋ ∈ V | |
2 | 1 | pwex 5281 | . . 3 ⊢ 𝒫 ℋ ∈ V |
3 | 2 | elpw2 5248 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ ↔ 𝐴 ⊆ 𝒫 ℋ) |
4 | unieq 4849 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
5 | 4 | fveq2d 6674 | . . . 4 ⊢ (𝑥 = 𝐴 → (⊥‘∪ 𝑥) = (⊥‘∪ 𝐴)) |
6 | 5 | fveq2d 6674 | . . 3 ⊢ (𝑥 = 𝐴 → (⊥‘(⊥‘∪ 𝑥)) = (⊥‘(⊥‘∪ 𝐴))) |
7 | df-chsup 29088 | . . 3 ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥))) | |
8 | fvex 6683 | . . 3 ⊢ (⊥‘(⊥‘∪ 𝐴)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6768 | . 2 ⊢ (𝐴 ∈ 𝒫 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) |
10 | 3, 9 | sylbir 237 | 1 ⊢ (𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘𝐴) = (⊥‘(⊥‘∪ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 ‘cfv 6355 ℋchba 28696 ⊥cort 28707 ∨ℋ chsup 28711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-hilex 28776 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-chsup 29088 |
This theorem is referenced by: chsupval 29112 hsupcl 29116 hsupss 29118 hsupunss 29120 sshjval3 29131 hsupval2 29186 |
Copyright terms: Public domain | W3C validator |