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Theorem cnmbfm 29158
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 2471 . . . . 5  |-  U. J  =  U. J
3 eqid 2471 . . . . 5  |-  U. K  =  U. K
42, 3cnf 20339 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
51, 4syl 17 . . 3  |-  ( ph  ->  F : U. J --> U. K )
6 cnmbfm.2 . . . . . 6  |-  ( ph  ->  S  =  (sigaGen `  J
) )
76unieqd 4200 . . . . 5  |-  ( ph  ->  U. S  =  U. (sigaGen `  J ) )
8 cntop1 20333 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
9 unisg 29039 . . . . . 6  |-  ( J  e.  Top  ->  U. (sigaGen `  J )  =  U. J )
101, 8, 93syl 18 . . . . 5  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
117, 10eqtrd 2505 . . . 4  |-  ( ph  ->  U. S  =  U. J )
12 cnmbfm.3 . . . . . 6  |-  ( ph  ->  T  =  (sigaGen `  K
) )
1312unieqd 4200 . . . . 5  |-  ( ph  ->  U. T  =  U. (sigaGen `  K ) )
14 cntop2 20334 . . . . . 6  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
15 unisg 29039 . . . . . 6  |-  ( K  e.  Top  ->  U. (sigaGen `  K )  =  U. K )
161, 14, 153syl 18 . . . . 5  |-  ( ph  ->  U. (sigaGen `  K
)  =  U. K
)
1713, 16eqtrd 2505 . . . 4  |-  ( ph  ->  U. T  =  U. K )
1811, 17feq23d 5734 . . 3  |-  ( ph  ->  ( F : U. S
--> U. T  <->  F : U. J --> U. K ) )
195, 18mpbird 240 . 2  |-  ( ph  ->  F : U. S --> U. T )
20 sssigagen 29041 . . . . . . 7  |-  ( J  e.  Top  ->  J  C_  (sigaGen `  J )
)
211, 8, 203syl 18 . . . . . 6  |-  ( ph  ->  J  C_  (sigaGen `  J
) )
2221, 6sseqtr4d 3455 . . . . 5  |-  ( ph  ->  J  C_  S )
2322adantr 472 . . . 4  |-  ( (
ph  /\  a  e.  K )  ->  J  C_  S )
24 cnima 20358 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  a  e.  K )  ->  ( `' F "
a )  e.  J
)
251, 24sylan 479 . . . 4  |-  ( (
ph  /\  a  e.  K )  ->  ( `' F " a )  e.  J )
2623, 25sseldd 3419 . . 3  |-  ( (
ph  /\  a  e.  K )  ->  ( `' F " a )  e.  S )
2726ralrimiva 2809 . 2  |-  ( ph  ->  A. a  e.  K  ( `' F " a )  e.  S )
28 elex 3040 . . . 4  |-  ( K  e.  Top  ->  K  e.  _V )
291, 14, 283syl 18 . . 3  |-  ( ph  ->  K  e.  _V )
30 sigagensiga 29037 . . . . . 6  |-  ( J  e.  Top  ->  (sigaGen `  J )  e.  (sigAlgebra ` 
U. J ) )
311, 8, 303syl 18 . . . . 5  |-  ( ph  ->  (sigaGen `  J )  e.  (sigAlgebra `  U. J ) )
326, 31eqeltrd 2549 . . . 4  |-  ( ph  ->  S  e.  (sigAlgebra `  U. J ) )
33 elrnsiga 29022 . . . 4  |-  ( S  e.  (sigAlgebra `  U. J )  ->  S  e.  U. ran sigAlgebra )
3432, 33syl 17 . . 3  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
3529, 34, 12imambfm 29157 . 2  |-  ( ph  ->  ( F  e.  ( SMblFnM T )  <->  ( F : U. S --> U. T  /\  A. a  e.  K  ( `' F " a )  e.  S ) ) )
3619, 27, 35mpbir2and 936 1  |-  ( ph  ->  F  e.  ( SMblFnM
T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    C_ wss 3390   U.cuni 4190   `'ccnv 4838   ran crn 4840   "cima 4842   -->wf 5585   ` cfv 5589  (class class class)co 6308   Topctop 19994    Cn ccn 20317  sigAlgebracsiga 29003  sigaGencsigagen 29034  MblFnMcmbfm 29145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-ac2 8911
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-oi 8043  df-card 8391  df-acn 8394  df-ac 8565  df-cda 8616  df-top 19998  df-topon 20000  df-cn 20320  df-siga 29004  df-sigagen 29035  df-mbfm 29146
This theorem is referenced by:  sxbrsiga  29185  rrvadd  29358  rrvmulc  29359
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