Theorem mbfmbfm 30865

Description: A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.)

Hypotheses

Ref	Expression
mbfmbfm.1	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
mbfmbfm.2	⊢ (𝜑 → 𝐽 ∈ Top)
mbfmbfm.3	⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))

Assertion

Ref	Expression
mbfmbfm	⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)

Proof of Theorem mbfmbfm

Step	Hyp	Ref	Expression
1		mbfmbfm.1	. . 3 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
2		measbasedom 30810	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
3	2	biimpi 208	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀))
4		measbase 30805	. . 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 30750	. 2 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
8		mbfmbfm.3	. 2 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))
9	5, 7, 8	isanmbfm 30863	1 ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)

Syntax hints:   → wi 4   ∈ wcel 2166  ∪ cuni 4658  dom cdm 5342  ran crn 5343  ‘cfv 6123  (class class class)co 6905  Topctop 21068  sigAlgebracsiga 30715  sigaGencsigagen 30746  measurescmeas 30803  MblFnMcmbfm 30857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-esum 30635  df-siga 30716  df-sigagen 30747  df-meas 30804  df-mbfm 30858

This theorem is referenced by: (None)