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Theorem cnmbfm 26614
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.
StepHypRef Expression
1 cnmbfm.1 . . . 4  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
2 eqid 2441 . . . . 5  |-  U. J  =  U. J
3 eqid 2441 . . . . 5  |-  U. K  =  U. K
42, 3cnf 18809 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
51, 4syl 16 . . 3  |-  ( ph  ->  F : U. J --> U. K )
6 cnmbfm.2 . . . . . 6  |-  ( ph  ->  S  =  (sigaGen `  J
) )
76unieqd 4098 . . . . 5  |-  ( ph  ->  U. S  =  U. (sigaGen `  J ) )
8 cntop1 18803 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
9 unisg 26522 . . . . . 6  |-  ( J  e.  Top  ->  U. (sigaGen `  J )  =  U. J )
101, 8, 93syl 20 . . . . 5  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
117, 10eqtrd 2473 . . . 4  |-  ( ph  ->  U. S  =  U. J )
12 cnmbfm.3 . . . . . 6  |-  ( ph  ->  T  =  (sigaGen `  K
) )
1312unieqd 4098 . . . . 5  |-  ( ph  ->  U. T  =  U. (sigaGen `  K ) )
14 cntop2 18804 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
15 unisg 26522 . . . . . 6  |-  ( K  e.  Top  ->  U. (sigaGen `  K )  =  U. K )
161, 14, 153syl 20 . . . . 5  |-  ( ph  ->  U. (sigaGen `  K
)  =  U. K
)
1713, 16eqtrd 2473 . . . 4  |-  ( ph  ->  U. T  =  U. K )
1811, 17feq23d 5551 . . 3  |-  ( ph  ->  ( F : U. S
--> U. T  <->  F : U. J --> U. K ) )
195, 18mpbird 232 . 2  |-  ( ph  ->  F : U. S --> U. T )
20 sssigagen 26524 . . . . . . 7  |-  ( J  e.  Top  ->  J  C_  (sigaGen `  J )
)
211, 8, 203syl 20 . . . . . 6  |-  ( ph  ->  J  C_  (sigaGen `  J
) )
2221, 6sseqtr4d 3390 . . . . 5  |-  ( ph  ->  J  C_  S )
2322adantr 462 . . . 4  |-  ( (
ph  /\  a  e.  K )  ->  J  C_  S )
24 cnima 18828 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  a  e.  K )  ->  ( `' F "
a )  e.  J
)
251, 24sylan 468 . . . 4  |-  ( (
ph  /\  a  e.  K )  ->  ( `' F " a )  e.  J )
2623, 25sseldd 3354 . . 3  |-  ( (
ph  /\  a  e.  K )  ->  ( `' F " a )  e.  S )
2726ralrimiva 2797 . 2  |-  ( ph  ->  A. a  e.  K  ( `' F " a )  e.  S )
28 elex 2979 . . . 4  |-  ( K  e.  Top  ->  K  e.  _V )
291, 14, 283syl 20 . . 3  |-  ( ph  ->  K  e.  _V )
30 sigagensiga 26520 . . . . . 6  |-  ( J  e.  Top  ->  (sigaGen `  J )  e.  (sigAlgebra ` 
U. J ) )
311, 8, 303syl 20 . . . . 5  |-  ( ph  ->  (sigaGen `  J )  e.  (sigAlgebra `  U. J ) )
326, 31eqeltrd 2515 . . . 4  |-  ( ph  ->  S  e.  (sigAlgebra `  U. J ) )
33 elrnsiga 26505 . . . 4  |-  ( S  e.  (sigAlgebra `  U. J )  ->  S  e.  U. ran sigAlgebra )
3432, 33syl 16 . . 3  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
3529, 34, 12imambfm 26613 . 2  |-  ( ph  ->  ( F  e.  ( SMblFnM T )  <->  ( F : U. S --> U. T  /\  A. a  e.  K  ( `' F " a )  e.  S ) ) )
3619, 27, 35mpbir2and 908 1  |-  ( ph  ->  F  e.  ( SMblFnM
T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    C_ wss 3325   U.cuni 4088   `'ccnv 4835   ran crn 4837   "cima 4839   -->wf 5411   ` cfv 5415  (class class class)co 6090   Topctop 18457    Cn ccn 18787  sigAlgebracsiga 26486  sigaGencsigagen 26517  MblFnMcmbfm 26601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-ac2 8628
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-oi 7720  df-card 8105  df-acn 8108  df-ac 8282  df-cda 8333  df-top 18462  df-topon 18465  df-cn 18790  df-siga 26487  df-sigagen 26518  df-mbfm 26602
This theorem is referenced by:  sxbrsiga  26641  rrvadd  26765  rrvmulc  26766
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