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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 31441 | . 2 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
2 | retop 23370 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
3 | df-sigagen 31398 | . . . . 5 ⊢ sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠}) | |
4 | 3 | funmpt2 6394 | . . . 4 ⊢ Fun sigaGen |
5 | fvex 6683 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ V | |
6 | sigagensiga 31400 | . . . . . 6 ⊢ ((topGen‘ran (,)) ∈ V → (sigaGen‘(topGen‘ran (,))) ∈ (sigAlgebra‘∪ (topGen‘ran (,)))) | |
7 | elrnsiga 31385 | . . . . . 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 31373 | . . . . 5 ⊢ ((sigaGen‘(topGen‘ran (,))) ∈ ∪ ran sigAlgebra → ∅ ∈ (sigaGen‘(topGen‘ran (,)))) | |
10 | elfvdm 6702 | . . . . 5 ⊢ (∅ ∈ (sigaGen‘(topGen‘ran (,))) → (topGen‘ran (,)) ∈ dom sigaGen) | |
11 | 8, 9, 10 | mp2b 10 | . . . 4 ⊢ (topGen‘ran (,)) ∈ dom sigaGen |
12 | funfvima 6992 | . . . 4 ⊢ ((Fun sigaGen ∧ (topGen‘ran (,)) ∈ dom sigaGen) → ((topGen‘ran (,)) ∈ Top → (sigaGen‘(topGen‘ran (,))) ∈ (sigaGen “ Top))) | |
13 | 4, 11, 12 | mp2an 690 | . . 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 2909 | 1 ⊢ 𝔅ℝ ∈ (sigaGen “ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 {crab 3142 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 ∪ cuni 4838 ∩ cint 4876 dom cdm 5555 ran crn 5556 “ cima 5558 Fun wfun 6349 ‘cfv 6355 (,)cioo 12739 topGenctg 16711 Topctop 21501 sigAlgebracsiga 31367 sigaGencsigagen 31397 𝔅ℝcbrsiga 31440 |
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-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 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-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 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-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-ioo 12743 df-topgen 16717 df-top 21502 df-bases 21554 df-siga 31368 df-sigagen 31398 df-brsiga 31441 |
This theorem is referenced by: (None) |
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