Theorem mbfmbfm 30320

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

Hypotheses

Ref	Expression
mbfmbfm.1	|- ( ph -> M e. U. ran measures )
mbfmbfm.2	|- ( ph -> J e. Top )
mbfmbfm.3	|- ( ph -> F e. ( dom MMblFnM (sigaGen ` J ) ) )

Assertion

Ref	Expression
mbfmbfm	|- ( ph -> F e. U. ran MblFnM )

Proof of Theorem mbfmbfm

Step	Hyp	Ref	Expression
1		mbfmbfm.1	. . 3 |- ( ph -> M e. U. ran measures )
2		measbasedom 30265	. . . 4 |- ( M e. U. ran measures <-> M e. (measures ` dom M ) )
3	2	biimpi 206	. . 3 |- ( M e. U. ran measures -> M e. (measures ` dom M ) )
4		measbase 30260	. . 3 |- ( M e. (measures ` dom M ) -> dom M e. U. ran sigAlgebra )
5	1, 3, 4	3syl 18	. 2 |- ( ph -> dom M e. U. ran sigAlgebra )
6		mbfmbfm.2	. . 3 |- ( ph -> J e. Top )
7	6	sgsiga 30205	. 2 |- ( ph -> (sigaGen ` J ) e. U. ran sigAlgebra )
8		mbfmbfm.3	. 2 |- ( ph -> F e. ( dom MMblFnM (sigaGen ` J ) ) )
9	5, 7, 8	isanmbfm 30318	1 |- ( ph -> F e. U. ran MblFnM )

Syntax hints:   -> wi 4   e. wcel 1990   U.cuni 4436   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650   Topctop 20698  sigAlgebracsiga 30170  sigaGencsigagen 30201  measurescmeas 30258  MblFnMcmbfm 30312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-esum 30090  df-siga 30171  df-sigagen 30202  df-meas 30259  df-mbfm 30313

This theorem is referenced by: (None)