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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 29572 | . 2 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
2 | retop 22375 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
3 | df-sigagen 29529 | . . . . 5 ⊢ sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠}) | |
4 | 3 | funmpt2 5841 | . . . 4 ⊢ Fun sigaGen |
5 | fvex 6113 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ V | |
6 | sigagensiga 29531 | . . . . . 6 ⊢ ((topGen‘ran (,)) ∈ V → (sigaGen‘(topGen‘ran (,))) ∈ (sigAlgebra‘∪ (topGen‘ran (,)))) | |
7 | elrnsiga 29516 | . . . . . 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 29504 | . . . . 5 ⊢ ((sigaGen‘(topGen‘ran (,))) ∈ ∪ ran sigAlgebra → ∅ ∈ (sigaGen‘(topGen‘ran (,)))) | |
10 | elfvdm 6130 | . . . . 5 ⊢ (∅ ∈ (sigaGen‘(topGen‘ran (,))) → (topGen‘ran (,)) ∈ dom sigaGen) | |
11 | 8, 9, 10 | mp2b 10 | . . . 4 ⊢ (topGen‘ran (,)) ∈ dom sigaGen |
12 | funfvima 6396 | . . . 4 ⊢ ((Fun sigaGen ∧ (topGen‘ran (,)) ∈ dom sigaGen) → ((topGen‘ran (,)) ∈ Top → (sigaGen‘(topGen‘ran (,))) ∈ (sigaGen “ Top))) | |
13 | 4, 11, 12 | mp2an 704 | . . 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 2684 | 1 ⊢ 𝔅ℝ ∈ (sigaGen “ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 ∩ cint 4410 dom cdm 5038 ran crn 5039 “ cima 5041 Fun wfun 5798 ‘cfv 5804 (,)cioo 12046 topGenctg 15921 Topctop 20517 sigAlgebracsiga 29497 sigaGencsigagen 29528 𝔅ℝcbrsiga 29571 |
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-8 1979 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-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ioo 12050 df-topgen 15927 df-top 20521 df-bases 20522 df-siga 29498 df-sigagen 29529 df-brsiga 29572 |
This theorem is referenced by: (None) |
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