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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 31461 | . . . 4 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀)) | |
3 | 2 | biimpi 218 | . . 3 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀)) |
4 | measbase 31456 | . . 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 31401 | . 2 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) |
8 | mbfmbfm.3 | . 2 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽))) | |
9 | 5, 7, 8 | isanmbfm 31514 | 1 ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∪ cuni 4838 dom cdm 5555 ran crn 5556 ‘cfv 6355 (class class class)co 7156 Topctop 21501 sigAlgebracsiga 31367 sigaGencsigagen 31397 measurescmeas 31454 MblFnMcmbfm 31508 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-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-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-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-esum 31287 df-siga 31368 df-sigagen 31398 df-meas 31455 df-mbfm 31509 |
This theorem is referenced by: (None) |
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