Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  orvcelval Unicode version

Theorem orvcelval 23684
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 23680 . 2  |-  ( ph  ->  ( XRV/𝑐  _E  A )  =  ( `' X " { x  e.  RR  |  x  _E  A } ) )
5 epelg 4322 . . . . . 6  |-  ( A  e. 𝔅  ->  ( x  _E  A  <->  x  e.  A ) )
63, 5syl 15 . . . . 5  |-  ( ph  ->  ( x  _E  A  <->  x  e.  A ) )
76rabbidv 2793 . . . 4  |-  ( ph  ->  { x  e.  RR  |  x  _E  A }  =  { x  e.  RR  |  x  e.  A } )
8 dfin5 3173 . . . . 5  |-  ( RR 
i^i  A )  =  { x  e.  RR  |  x  e.  A }
98a1i 10 . . . 4  |-  ( ph  ->  ( RR  i^i  A
)  =  { x  e.  RR  |  x  e.  A } )
10 elssuni 3871 . . . . . . 7  |-  ( A  e. 𝔅  ->  A  C_  U.𝔅
)
11 unibrsiga 23532 . . . . . . 7  |-  U.𝔅  =  RR
1210, 11syl6sseq 3237 . . . . . 6  |-  ( A  e. 𝔅  ->  A  C_  RR )
133, 12syl 15 . . . . 5  |-  ( ph  ->  A  C_  RR )
14 sseqin2 3401 . . . . 5  |-  ( A 
C_  RR  <->  ( RR  i^i  A )  =  A )
1513, 14sylib 188 . . . 4  |-  ( ph  ->  ( RR  i^i  A
)  =  A )
167, 9, 153eqtr2d 2334 . . 3  |-  ( ph  ->  { x  e.  RR  |  x  _E  A }  =  A )
1716imaeq2d 5028 . 2  |-  ( ph  ->  ( `' X " { x  e.  RR  |  x  _E  A } )  =  ( `' X " A ) )
184, 17eqtrd 2328 1  |-  ( ph  ->  ( XRV/𝑐  _E  A )  =  ( `' X " A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {crab 2560    i^i cin 3164    C_ wss 3165   U.cuni 3843   class class class wbr 4039    _E cep 4319   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874   RRcr 8752  𝔅cbrsiga 23527  Probcprb 23625  rRndVarcrrv 23658  ∘RV/𝑐corvc 23671
This theorem is referenced by:  orvcelel  23685  dstrvval  23686  dstrvprob  23687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-ioo 10676  df-topgen 13360  df-top 16652  df-bases 16654  df-esum 23426  df-siga 23484  df-sigagen 23515  df-brsiga 23528  df-meas 23542  df-mbfm 23571  df-prob 23626  df-rrv 23659  df-orvc 23672
  Copyright terms: Public domain W3C validator