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Theorem orvcelval 24728
Description: Preimage maps produced by the "elementhood" relation (Contributed by Thierry Arnoux, 6-Feb-2017.)
Hypotheses
Ref Expression
dstrvprob.1  |-  ( ph  ->  P  e. Prob )
dstrvprob.2  |-  ( ph  ->  X  e.  (rRndVar `  P
) )
orvcelel.1  |-  ( ph  ->  A  e. 𝔅 )
Assertion
Ref Expression
orvcelval  |-  ( ph  ->  ( XRV/𝑐  _E  A )  =  ( `' X " A ) )

Proof of Theorem orvcelval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dstrvprob.1 . . 3  |-  ( ph  ->  P  e. Prob )
2 dstrvprob.2 . . 3  |-  ( ph  ->  X  e.  (rRndVar `  P
) )
3 orvcelel.1 . . 3  |-  ( ph  ->  A  e. 𝔅 )
41, 2, 3orrvcval4 24724 . 2  |-  ( ph  ->  ( XRV/𝑐  _E  A )  =  ( `' X " { x  e.  RR  |  x  _E  A } ) )
5 epelg 4497 . . . . . 6  |-  ( A  e. 𝔅  ->  ( x  _E  A  <->  x  e.  A ) )
63, 5syl 16 . . . . 5  |-  ( ph  ->  ( x  _E  A  <->  x  e.  A ) )
76rabbidv 2950 . . . 4  |-  ( ph  ->  { x  e.  RR  |  x  _E  A }  =  { x  e.  RR  |  x  e.  A } )
8 dfin5 3330 . . . . 5  |-  ( RR 
i^i  A )  =  { x  e.  RR  |  x  e.  A }
98a1i 11 . . . 4  |-  ( ph  ->  ( RR  i^i  A
)  =  { x  e.  RR  |  x  e.  A } )
10 elssuni 4045 . . . . . . 7  |-  ( A  e. 𝔅  ->  A  C_  U.𝔅
)
11 unibrsiga 24542 . . . . . . 7  |-  U.𝔅  =  RR
1210, 11syl6sseq 3396 . . . . . 6  |-  ( A  e. 𝔅  ->  A  C_  RR )
133, 12syl 16 . . . . 5  |-  ( ph  ->  A  C_  RR )
14 sseqin2 3562 . . . . 5  |-  ( A 
C_  RR  <->  ( RR  i^i  A )  =  A )
1513, 14sylib 190 . . . 4  |-  ( ph  ->  ( RR  i^i  A
)  =  A )
167, 9, 153eqtr2d 2476 . . 3  |-  ( ph  ->  { x  e.  RR  |  x  _E  A }  =  A )
1716imaeq2d 5205 . 2  |-  ( ph  ->  ( `' X " { x  e.  RR  |  x  _E  A } )  =  ( `' X " A ) )
184, 17eqtrd 2470 1  |-  ( ph  ->  ( XRV/𝑐  _E  A )  =  ( `' X " A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   {crab 2711    i^i cin 3321    C_ wss 3322   U.cuni 4017   class class class wbr 4214    _E cep 4494   `'ccnv 4879   "cima 4883   ` cfv 5456  (class class class)co 6083   RRcr 8991  𝔅cbrsiga 24537  Probcprb 24667  rRndVarcrrv 24700  ∘RV/𝑐corvc 24715
This theorem is referenced by:  orvcelel  24729  dstrvval  24730  dstrvprob  24731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-pre-lttri 9066  ax-pre-lttrn 9067
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-ioo 10922  df-topgen 13669  df-top 16965  df-bases 16967  df-esum 24427  df-siga 24493  df-sigagen 24524  df-brsiga 24538  df-meas 24552  df-mbfm 24603  df-prob 24668  df-rrv 24701  df-orvc 24716
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