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Mirrors > Home > MPE Home > Th. List > Mathboxes > brsiga | Structured version Visualization version GIF version |
Description: The Borel Algebra on real numbers is a Borel sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.) |
Ref | Expression |
---|---|
brsiga | ⊢ 𝔅ℝ ∈ (sigaGen “ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-brsiga 30525 | . 2 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
2 | retop 22737 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
3 | df-sigagen 30482 | . . . . 5 ⊢ sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠}) | |
4 | 3 | funmpt2 6076 | . . . 4 ⊢ Fun sigaGen |
5 | fvex 6350 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ V | |
6 | sigagensiga 30484 | . . . . . 6 ⊢ ((topGen‘ran (,)) ∈ V → (sigaGen‘(topGen‘ran (,))) ∈ (sigAlgebra‘∪ (topGen‘ran (,)))) | |
7 | elrnsiga 30469 | . . . . . 6 ⊢ ((sigaGen‘(topGen‘ran (,))) ∈ (sigAlgebra‘∪ (topGen‘ran (,))) → (sigaGen‘(topGen‘ran (,))) ∈ ∪ ran sigAlgebra) | |
8 | 5, 6, 7 | mp2b 10 | . . . . 5 ⊢ (sigaGen‘(topGen‘ran (,))) ∈ ∪ ran sigAlgebra |
9 | 0elsiga 30457 | . . . . 5 ⊢ ((sigaGen‘(topGen‘ran (,))) ∈ ∪ ran sigAlgebra → ∅ ∈ (sigaGen‘(topGen‘ran (,)))) | |
10 | elfvdm 6369 | . . . . 5 ⊢ (∅ ∈ (sigaGen‘(topGen‘ran (,))) → (topGen‘ran (,)) ∈ dom sigaGen) | |
11 | 8, 9, 10 | mp2b 10 | . . . 4 ⊢ (topGen‘ran (,)) ∈ dom sigaGen |
12 | funfvima 6643 | . . . 4 ⊢ ((Fun sigaGen ∧ (topGen‘ran (,)) ∈ dom sigaGen) → ((topGen‘ran (,)) ∈ Top → (sigaGen‘(topGen‘ran (,))) ∈ (sigaGen “ Top))) | |
13 | 4, 11, 12 | mp2an 710 | . . 3 ⊢ ((topGen‘ran (,)) ∈ Top → (sigaGen‘(topGen‘ran (,))) ∈ (sigaGen “ Top)) |
14 | 2, 13 | ax-mp 5 | . 2 ⊢ (sigaGen‘(topGen‘ran (,))) ∈ (sigaGen “ Top) |
15 | 1, 14 | eqeltri 2823 | 1 ⊢ 𝔅ℝ ∈ (sigaGen “ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2127 {crab 3042 Vcvv 3328 ⊆ wss 3703 ∅c0 4046 ∪ cuni 4576 ∩ cint 4615 dom cdm 5254 ran crn 5255 “ cima 5257 Fun wfun 6031 ‘cfv 6037 (,)cioo 12339 topGenctg 16271 Topctop 20871 sigAlgebracsiga 30450 sigaGencsigagen 30481 𝔅ℝcbrsiga 30524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-pre-lttri 10173 ax-pre-lttrn 10174 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-fal 1626 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-id 5162 df-po 5175 df-so 5176 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-1st 7321 df-2nd 7322 df-er 7899 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-ioo 12343 df-topgen 16277 df-top 20872 df-bases 20923 df-siga 30451 df-sigagen 30482 df-brsiga 30525 |
This theorem is referenced by: (None) |
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