Theorem mbfmbfm 31520

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 31465	. . . 4 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
3	2	biimpi 218	. . 3 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀))
4		measbase 31460	. . 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 31405	. 2 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
8		mbfmbfm.3	. 2 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))
9	5, 7, 8	isanmbfm 31518	1 ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)

Syntax hints:   → wi 4   ∈ wcel 2113  ∪ cuni 4841  dom cdm 5558  ran crn 5559  ‘cfv 6358  (class class class)co 7159  Topctop 21504  sigAlgebracsiga 31371  sigaGencsigagen 31401  measurescmeas 31458  MblFnMcmbfm 31512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-esum 31291  df-siga 31372  df-sigagen 31402  df-meas 31459  df-mbfm 31513

This theorem is referenced by: (None)