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Theorem orvcelval 28671
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 28667 . 2  |-  ( ph  ->  ( XRV/𝑐  _E  A )  =  ( `' X " { x  e.  RR  |  x  _E  A } ) )
5 epelg 4781 . . . . . 6  |-  ( A  e. 𝔅  ->  ( x  _E  A  <->  x  e.  A ) )
63, 5syl 16 . . . . 5  |-  ( ph  ->  ( x  _E  A  <->  x  e.  A ) )
76rabbidv 3098 . . . 4  |-  ( ph  ->  { x  e.  RR  |  x  _E  A }  =  { x  e.  RR  |  x  e.  A } )
8 dfin5 3469 . . . . 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 4264 . . . . . . 7  |-  ( A  e. 𝔅  ->  A  C_  U.𝔅
)
11 unibrsiga 28394 . . . . . . 7  |-  U.𝔅  =  RR
1210, 11syl6sseq 3535 . . . . . 6  |-  ( A  e. 𝔅  ->  A  C_  RR )
133, 12syl 16 . . . . 5  |-  ( ph  ->  A  C_  RR )
14 sseqin2 3703 . . . . 5  |-  ( A 
C_  RR  <->  ( RR  i^i  A )  =  A )
1513, 14sylib 196 . . . 4  |-  ( ph  ->  ( RR  i^i  A
)  =  A )
167, 9, 153eqtr2d 2501 . . 3  |-  ( ph  ->  { x  e.  RR  |  x  _E  A }  =  A )
1716imaeq2d 5325 . 2  |-  ( ph  ->  ( `' X " { x  e.  RR  |  x  _E  A } )  =  ( `' X " A ) )
184, 17eqtrd 2495 1  |-  ( ph  ->  ( XRV/𝑐  _E  A )  =  ( `' X " A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823   {crab 2808    i^i cin 3460    C_ wss 3461   U.cuni 4235   class class class wbr 4439    _E cep 4778   `'ccnv 4987   "cima 4991   ` cfv 5570  (class class class)co 6270   RRcr 9480  𝔅cbrsiga 28389  Probcprb 28610  rRndVarcrrv 28643  ∘RV/𝑐corvc 28658
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-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  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-nel 2652  df-ral 2809  df-rex 2810  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-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  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-1st 6773  df-2nd 6774  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-ioo 11536  df-topgen 14933  df-top 19566  df-bases 19568  df-esum 28257  df-siga 28338  df-sigagen 28369  df-brsiga 28390  df-meas 28404  df-mbfm 28459  df-prob 28611  df-rrv 28644  df-orvc 28659
This theorem is referenced by:  orvcelel  28672  dstrvval  28673  dstrvprob  28674
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