Theorem cnmbfm 30325

Description: A continuous function is measurable with respect to the Borel Algebra of its domain and range. (Contributed by Thierry Arnoux, 3-Jun-2017.)

Hypotheses

Ref	Expression
cnmbfm.1	|- ( ph -> F e. ( J Cn K ) )
cnmbfm.2	|- ( ph -> S = (sigaGen ` J ) )
cnmbfm.3	|- ( ph -> T = (sigaGen ` K ) )

Assertion

Ref	Expression
cnmbfm	|- ( ph -> F e. ( SMblFnM T ) )

Proof of Theorem cnmbfm

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnmbfm.1	. . . 4 |- ( ph -> F e. ( J Cn K ) )
2		eqid 2622	. . . . 5 |- U. J = U. J
3		eqid 2622	. . . . 5 |- U. K = U. K
4	2, 3	cnf 21050	. . . 4 |- ( F e. ( J Cn K ) -> F : U. J --> U. K )
5	1, 4	syl 17	. . 3 |- ( ph -> F : U. J --> U. K )
6		cnmbfm.2	. . . . . 6 |- ( ph -> S = (sigaGen ` J ) )
7	6	unieqd 4446	. . . . 5 |- ( ph -> U. S = U. (sigaGen ` J ) )
8		cntop1 21044	. . . . . 6 |- ( F e. ( J Cn K ) -> J e. Top )
9		unisg 30206	. . . . . 6 |- ( J e. Top -> U. (sigaGen ` J ) = U. J )
10	1, 8, 9	3syl 18	. . . . 5 |- ( ph -> U. (sigaGen ` J ) = U. J )
11	7, 10	eqtrd 2656	. . . 4 |- ( ph -> U. S = U. J )
12		cnmbfm.3	. . . . . 6 |- ( ph -> T = (sigaGen ` K ) )
13	12	unieqd 4446	. . . . 5 |- ( ph -> U. T = U. (sigaGen ` K ) )
14		cntop2 21045	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
15		unisg 30206	. . . . . 6 |- ( K e. Top -> U. (sigaGen ` K ) = U. K )
16	1, 14, 15	3syl 18	. . . . 5 |- ( ph -> U. (sigaGen ` K ) = U. K )
17	13, 16	eqtrd 2656	. . . 4 |- ( ph -> U. T = U. K )
18	11, 17	feq23d 6040	. . 3 |- ( ph -> ( F : U. S --> U. T <-> F : U. J --> U. K ) )
19	5, 18	mpbird 247	. 2 |- ( ph -> F : U. S --> U. T )
20		sssigagen 30208	. . . . . . 7 |- ( J e. Top -> J C_ (sigaGen ` J ) )
21	1, 8, 20	3syl 18	. . . . . 6 |- ( ph -> J C_ (sigaGen ` J ) )
22	21, 6	sseqtr4d 3642	. . . . 5 |- ( ph -> J C_ S )
23	22	adantr 481	. . . 4 |- ( ( ph /\ a e. K ) -> J C_ S )
24		cnima 21069	. . . . 5 |- ( ( F e. ( J Cn K ) /\ a e. K ) -> ( `' F " a ) e. J )
25	1, 24	sylan 488	. . . 4 |- ( ( ph /\ a e. K ) -> ( `' F " a ) e. J )
26	23, 25	sseldd 3604	. . 3 |- ( ( ph /\ a e. K ) -> ( `' F " a ) e. S )
27	26	ralrimiva 2966	. 2 |- ( ph -> A. a e. K ( `' F " a ) e. S )
28		elex 3212	. . . 4 |- ( K e. Top -> K e. _V )
29	1, 14, 28	3syl 18	. . 3 |- ( ph -> K e. _V )
30		sigagensiga 30204	. . . . . 6 |- ( J e. Top -> (sigaGen ` J ) e. (sigAlgebra ` U. J ) )
31	1, 8, 30	3syl 18	. . . . 5 |- ( ph -> (sigaGen ` J ) e. (sigAlgebra ` U. J ) )
32	6, 31	eqeltrd 2701	. . . 4 |- ( ph -> S e. (sigAlgebra ` U. J ) )
33		elrnsiga 30189	. . . 4 |- ( S e. (sigAlgebra ` U. J ) -> S e. U. ran sigAlgebra )
34	32, 33	syl 17	. . 3 |- ( ph -> S e. U. ran sigAlgebra )
35	29, 34, 12	imambfm 30324	. 2 |- ( ph -> ( F e. ( SMblFnM T ) <-> ( F : U. S --> U. T /\ A. a e. K ( `' F " a ) e. S ) ) )
36	19, 27, 35	mpbir2and 957	1 |- ( ph -> F e. ( SMblFnM T ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   U.cuni 4436   `'ccnv 5113   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   Topctop 20698   Cn ccn 21028  sigAlgebracsiga 30170  sigaGencsigagen 30201  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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-top 20699  df-topon 20716  df-cn 21031  df-siga 30171  df-sigagen 30202  df-mbfm 30313

This theorem is referenced by:  sxbrsiga  30352  rrvadd  30514  rrvmulc  30515