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Theorem sibfima 24606
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 24601 . . . 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 202 . . 3  |-  ( ph  ->  ( F  e.  ( dom  MMblFnM S )  /\  ran  F  e.  Fin  /\  A. x  e.  ( ran 
F  \  {  .0.  } ) ( M `  ( `' F " { x } ) )  e.  ( 0 [,)  +oo ) ) )
1211simp3d 971 . 2  |-  ( ph  ->  A. x  e.  ( ran  F  \  {  .0.  } ) ( M `
 ( `' F " { x } ) )  e.  ( 0 [,)  +oo ) )
13 sneq 3785 . . . . . 6  |-  ( x  =  A  ->  { x }  =  { A } )
1413imaeq2d 5162 . . . . 5  |-  ( x  =  A  ->  ( `' F " { x } )  =  ( `' F " { A } ) )
1514fveq2d 5691 . . . 4  |-  ( x  =  A  ->  ( M `  ( `' F " { x }
) )  =  ( M `  ( `' F " { A } ) ) )
1615eleq1d 2470 . . 3  |-  ( x  =  A  ->  (
( M `  ( `' F " { x } ) )  e.  ( 0 [,)  +oo ) 
<->  ( M `  ( `' F " { A } ) )  e.  ( 0 [,)  +oo ) ) )
1716rspcv 3008 . 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 456 1  |-  ( (
ph  /\  A  e.  ( ran  F  \  {  .0.  } ) )  -> 
( M `  ( `' F " { A } ) )  e.  ( 0 [,)  +oo ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666    \ cdif 3277   {csn 3774   U.cuni 3975   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   ` cfv 5413  (class class class)co 6040   Fincfn 7068   0cc0 8946    +oocpnf 9073   [,)cico 10874   Basecbs 13424  Scalarcsca 13487   .scvsca 13488   TopOpenctopn 13604   0gc0g 13678  RRHomcrrh 24330  sigaGencsigagen 24474  measurescmeas 24502  MblFnMcmbfm 24553  sitgcsitg 24597
This theorem is referenced by:  sibfof  24607  sitgfval  24608  sitgclg  24609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-sitg 24598
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