Theorem mbfmco 30326

Description: The composition of two measurable functions is measurable. ( cf. cnmpt11 21466) (Contributed by Thierry Arnoux, 4-Jun-2017.)

Hypotheses

Ref	Expression
mbfmco.1	|- ( ph -> R e. U. ran sigAlgebra )
mbfmco.2	|- ( ph -> S e. U. ran sigAlgebra )
mbfmco.3	|- ( ph -> T e. U. ran sigAlgebra )
mbfmco.4	|- ( ph -> F e. ( RMblFnM S ) )
mbfmco.5	|- ( ph -> G e. ( SMblFnM T ) )

Assertion

Ref	Expression
mbfmco	|- ( ph -> ( G o. F ) e. ( RMblFnM T ) )

Proof of Theorem mbfmco

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfmco.2	. . . . 5 |- ( ph -> S e. U. ran sigAlgebra )
2		mbfmco.3	. . . . 5 |- ( ph -> T e. U. ran sigAlgebra )
3		mbfmco.5	. . . . 5 |- ( ph -> G e. ( SMblFnM T ) )
4	1, 2, 3	mbfmf 30317	. . . 4 |- ( ph -> G : U. S --> U. T )
5		mbfmco.1	. . . . 5 |- ( ph -> R e. U. ran sigAlgebra )
6		mbfmco.4	. . . . 5 |- ( ph -> F e. ( RMblFnM S ) )
7	5, 1, 6	mbfmf 30317	. . . 4 |- ( ph -> F : U. R --> U. S )
8		fco 6058	. . . 4 |- ( ( G : U. S --> U. T /\ F : U. R --> U. S ) -> ( G o. F ) : U. R --> U. T )
9	4, 7, 8	syl2anc 693	. . 3 |- ( ph -> ( G o. F ) : U. R --> U. T )
10		unielsiga 30191	. . . . 5 |- ( T e. U. ran sigAlgebra -> U. T e. T )
11	2, 10	syl 17	. . . 4 |- ( ph -> U. T e. T )
12		unielsiga 30191	. . . . 5 |- ( R e. U. ran sigAlgebra -> U. R e. R )
13	5, 12	syl 17	. . . 4 |- ( ph -> U. R e. R )
14	11, 13	elmapd 7871	. . 3 |- ( ph -> ( ( G o. F ) e. ( U. T ^m U. R ) <-> ( G o. F ) : U. R --> U. T ) )
15	9, 14	mpbird 247	. 2 |- ( ph -> ( G o. F ) e. ( U. T ^m U. R ) )
16		cnvco 5308	. . . . . 6 |- `' ( G o. F ) = ( `' F o. `' G )
17	16	imaeq1i 5463	. . . . 5 |- ( `' ( G o. F ) " a ) = ( ( `' F o. `' G ) " a )
18		imaco 5640	. . . . 5 |- ( ( `' F o. `' G ) " a ) = ( `' F " ( `' G " a ) )
19	17, 18	eqtri 2644	. . . 4 |- ( `' ( G o. F ) " a ) = ( `' F " ( `' G " a ) )
20	5	adantr 481	. . . . 5 |- ( ( ph /\ a e. T ) -> R e. U. ran sigAlgebra )
21	1	adantr 481	. . . . 5 |- ( ( ph /\ a e. T ) -> S e. U. ran sigAlgebra )
22	6	adantr 481	. . . . 5 |- ( ( ph /\ a e. T ) -> F e. ( RMblFnM S ) )
23	2	adantr 481	. . . . . 6 |- ( ( ph /\ a e. T ) -> T e. U. ran sigAlgebra )
24	3	adantr 481	. . . . . 6 |- ( ( ph /\ a e. T ) -> G e. ( SMblFnM T ) )
25		simpr 477	. . . . . 6 |- ( ( ph /\ a e. T ) -> a e. T )
26	21, 23, 24, 25	mbfmcnvima 30319	. . . . 5 |- ( ( ph /\ a e. T ) -> ( `' G " a ) e. S )
27	20, 21, 22, 26	mbfmcnvima 30319	. . . 4 |- ( ( ph /\ a e. T ) -> ( `' F " ( `' G " a ) ) e. R )
28	19, 27	syl5eqel 2705	. . 3 |- ( ( ph /\ a e. T ) -> ( `' ( G o. F ) " a ) e. R )
29	28	ralrimiva 2966	. 2 |- ( ph -> A. a e. T ( `' ( G o. F ) " a ) e. R )
30	5, 2	ismbfm 30314	. 2 |- ( ph -> ( ( G o. F ) e. ( RMblFnM T ) <-> ( ( G o. F ) e. ( U. T ^m U. R ) /\ A. a e. T ( `' ( G o. F ) " a ) e. R ) ) )
31	15, 29, 30	mpbir2and 957	1 |- ( ph -> ( G o. F ) e. ( RMblFnM T ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   U.cuni 4436   `'ccnv 5113   ran crn 5115   "cima 5117   o. ccom 5118   -->wf 5884  (class class class)co 6650   ^m cmap 7857  sigAlgebracsiga 30170  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-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-map 7859  df-siga 30171  df-mbfm 30313

This theorem is referenced by:  rrvadd  30514  rrvmulc  30515