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Theorem sibfima 28544
Description: Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
Hypotheses
Ref Expression
sitgval.b  |-  B  =  ( Base `  W
)
sitgval.j  |-  J  =  ( TopOpen `  W )
sitgval.s  |-  S  =  (sigaGen `  J )
sitgval.0  |-  .0.  =  ( 0g `  W )
sitgval.x  |-  .x.  =  ( .s `  W )
sitgval.h  |-  H  =  (RRHom `  (Scalar `  W
) )
sitgval.1  |-  ( ph  ->  W  e.  V )
sitgval.2  |-  ( ph  ->  M  e.  U. ran measures )
sibfmbl.1  |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )
Assertion
Ref Expression
sibfima  |-  ( (
ph  /\  A  e.  ( ran  F  \  {  .0.  } ) )  -> 
( M `  ( `' F " { A } ) )  e.  ( 0 [,) +oo ) )

Proof of Theorem sibfima
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sibfmbl.1 . . . 4  |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )
2 sitgval.b . . . . 5  |-  B  =  ( Base `  W
)
3 sitgval.j . . . . 5  |-  J  =  ( TopOpen `  W )
4 sitgval.s . . . . 5  |-  S  =  (sigaGen `  J )
5 sitgval.0 . . . . 5  |-  .0.  =  ( 0g `  W )
6 sitgval.x . . . . 5  |-  .x.  =  ( .s `  W )
7 sitgval.h . . . . 5  |-  H  =  (RRHom `  (Scalar `  W
) )
8 sitgval.1 . . . . 5  |-  ( ph  ->  W  e.  V )
9 sitgval.2 . . . . 5  |-  ( ph  ->  M  e.  U. ran measures )
102, 3, 4, 5, 6, 7, 8, 9issibf 28539 . . . 4  |-  ( ph  ->  ( F  e.  dom  ( Wsitg M )  <->  ( F  e.  ( dom  MMblFnM S
)  /\  ran  F  e. 
Fin  /\  A. x  e.  ( ran  F  \  {  .0.  } ) ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) ) ) )
111, 10mpbid 210 . . 3  |-  ( ph  ->  ( F  e.  ( dom  MMblFnM S )  /\  ran  F  e.  Fin  /\  A. x  e.  ( ran 
F  \  {  .0.  } ) ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) ) )
1211simp3d 1008 . 2  |-  ( ph  ->  A. x  e.  ( ran  F  \  {  .0.  } ) ( M `
 ( `' F " { x } ) )  e.  ( 0 [,) +oo ) )
13 sneq 4026 . . . . . 6  |-  ( x  =  A  ->  { x }  =  { A } )
1413imaeq2d 5325 . . . . 5  |-  ( x  =  A  ->  ( `' F " { x } )  =  ( `' F " { A } ) )
1514fveq2d 5852 . . . 4  |-  ( x  =  A  ->  ( M `  ( `' F " { x }
) )  =  ( M `  ( `' F " { A } ) ) )
1615eleq1d 2523 . . 3  |-  ( x  =  A  ->  (
( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) 
<->  ( M `  ( `' F " { A } ) )  e.  ( 0 [,) +oo ) ) )
1716rspcv 3203 . 2  |-  ( A  e.  ( ran  F  \  {  .0.  } )  ->  ( A. x  e.  ( ran  F  \  {  .0.  } ) ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo )  ->  ( M `  ( `' F " { A } ) )  e.  ( 0 [,) +oo ) ) )
1812, 17mpan9 467 1  |-  ( (
ph  /\  A  e.  ( ran  F  \  {  .0.  } ) )  -> 
( M `  ( `' F " { A } ) )  e.  ( 0 [,) +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804    \ cdif 3458   {csn 4016   U.cuni 4235   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991   ` cfv 5570  (class class class)co 6270   Fincfn 7509   0cc0 9481   +oocpnf 9614   [,)cico 11534   Basecbs 14716  Scalarcsca 14787   .scvsca 14788   TopOpenctopn 14911   0gc0g 14929  RRHomcrrh 28208  sigaGencsigagen 28368  measurescmeas 28403  MblFnMcmbfm 28458  sitgcsitg 28535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-sitg 28536
This theorem is referenced by:  sibfinima  28545  sitgfval  28547  sitgclg  28548
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