![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mbfmbfm | Structured version Visualization version GIF version |
Description: A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.) |
Ref | Expression |
---|---|
mbfmbfm.1 | ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) |
mbfmbfm.2 | ⊢ (𝜑 → 𝐽 ∈ Top) |
mbfmbfm.3 | ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽))) |
Ref | Expression |
---|---|
mbfmbfm | ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfmbfm.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) | |
2 | measbasedom 31571 | . . . 4 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀)) | |
3 | 2 | biimpi 219 | . . 3 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀)) |
4 | measbase 31566 | . . 3 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → dom 𝑀 ∈ ∪ ran sigAlgebra) | |
5 | 1, 3, 4 | 3syl 18 | . 2 ⊢ (𝜑 → dom 𝑀 ∈ ∪ ran sigAlgebra) |
6 | mbfmbfm.2 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) | |
7 | 6 | sgsiga 31511 | . 2 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
8 | mbfmbfm.3 | . 2 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽))) | |
9 | 5, 7, 8 | isanmbfm 31624 | 1 ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∪ cuni 4800 dom cdm 5519 ran crn 5520 ‘cfv 6324 (class class class)co 7135 Topctop 21498 sigAlgebracsiga 31477 sigaGencsigagen 31507 measurescmeas 31564 MblFnMcmbfm 31618 |
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-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 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-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 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-int 4839 df-iun 4883 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-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-esum 31397 df-siga 31478 df-sigagen 31508 df-meas 31565 df-mbfm 31619 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |