Theorem mbfmbfm 29647

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 29592	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
3	2	biimpi 205	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀))
4		measbase 29587	. . 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 29532	. 2 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
8		mbfmbfm.3	. 2 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))
9	5, 7, 8	isanmbfm 29645	1 ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)

Syntax hints:   → wi 4   ∈ wcel 1977  ∪ cuni 4372  dom cdm 5038  ran crn 5039  ‘cfv 5804  (class class class)co 6549  Topctop 20517  sigAlgebracsiga 29497  sigaGencsigagen 29528  measurescmeas 29585  MblFnMcmbfm 29639

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

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-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-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-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-esum 29417  df-siga 29498  df-sigagen 29529  df-meas 29586  df-mbfm 29640

This theorem is referenced by: (None)