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Theorem orvcval 28619
Description: Value of the preimage mapping operator applied on a given random variable and constant (Contributed by Thierry Arnoux, 19-Jan-2017.)
Hypotheses
Ref Expression
orvcval.1  |-  ( ph  ->  Fun  X )
orvcval.2  |-  ( ph  ->  X  e.  V )
orvcval.3  |-  ( ph  ->  A  e.  W )
Assertion
Ref Expression
orvcval  |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  |  y R A } ) )
Distinct variable groups:    y, A    y, R    y, X
Allowed substitution hints:    ph( y)    V( y)    W( y)

Proof of Theorem orvcval
Dummy variables  x  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-orvc 28618 . . 3  |-RV/𝑐 R  =  ( x  e.  { x  |  Fun  x } , 
a  e.  _V  |->  ( `' x " { y  |  y R a } ) )
21a1i 11 . 2  |-  ( ph  ->RV/𝑐 R  =  ( x  e. 
{ x  |  Fun  x } ,  a  e. 
_V  |->  ( `' x " { y  |  y R a } ) ) )
3 simpl 455 . . . . 5  |-  ( ( x  =  X  /\  a  =  A )  ->  x  =  X )
43cnveqd 5108 . . . 4  |-  ( ( x  =  X  /\  a  =  A )  ->  `' x  =  `' X )
5 simpr 459 . . . . . 6  |-  ( ( x  =  X  /\  a  =  A )  ->  a  =  A )
65breq2d 4396 . . . . 5  |-  ( ( x  =  X  /\  a  =  A )  ->  ( y R a  <-> 
y R A ) )
76abbidv 2532 . . . 4  |-  ( ( x  =  X  /\  a  =  A )  ->  { y  |  y R a }  =  { y  |  y R A } )
84, 7imaeq12d 5267 . . 3  |-  ( ( x  =  X  /\  a  =  A )  ->  ( `' x " { y  |  y R a } )  =  ( `' X " { y  |  y R A } ) )
98adantl 464 . 2  |-  ( (
ph  /\  ( x  =  X  /\  a  =  A ) )  -> 
( `' x " { y  |  y R a } )  =  ( `' X " { y  |  y R A } ) )
10 orvcval.1 . . 3  |-  ( ph  ->  Fun  X )
11 orvcval.2 . . . 4  |-  ( ph  ->  X  e.  V )
12 funeq 5532 . . . . 5  |-  ( x  =  X  ->  ( Fun  x  <->  Fun  X ) )
1312elabg 3189 . . . 4  |-  ( X  e.  V  ->  ( X  e.  { x  |  Fun  x }  <->  Fun  X ) )
1411, 13syl 16 . . 3  |-  ( ph  ->  ( X  e.  {
x  |  Fun  x } 
<->  Fun  X ) )
1510, 14mpbird 232 . 2  |-  ( ph  ->  X  e.  { x  |  Fun  x } )
16 orvcval.3 . . 3  |-  ( ph  ->  A  e.  W )
17 elex 3060 . . 3  |-  ( A  e.  W  ->  A  e.  _V )
1816, 17syl 16 . 2  |-  ( ph  ->  A  e.  _V )
19 cnvexg 6667 . . 3  |-  ( X  e.  V  ->  `' X  e.  _V )
20 imaexg 6658 . . 3  |-  ( `' X  e.  _V  ->  ( `' X " { y  |  y R A } )  e.  _V )
2111, 19, 203syl 20 . 2  |-  ( ph  ->  ( `' X " { y  |  y R A } )  e.  _V )
222, 9, 15, 18, 21ovmpt2d 6351 1  |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  |  y R A } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836   {cab 2381   _Vcvv 3051   class class class wbr 4384   `'ccnv 4929   "cima 4933   Fun wfun 5507  (class class class)co 6218    |-> cmpt2 6220  ∘RV/𝑐corvc 28617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-orvc 28618
This theorem is referenced by:  orvcval2  28620  orvcval4  28622
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