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Theorem orvcval 27004
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 27003 . . 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 457 . . . . 5  |-  ( ( x  =  X  /\  a  =  A )  ->  x  =  X )
43cnveqd 5126 . . . 4  |-  ( ( x  =  X  /\  a  =  A )  ->  `' x  =  `' X )
5 simpr 461 . . . . . 6  |-  ( ( x  =  X  /\  a  =  A )  ->  a  =  A )
65breq2d 4415 . . . . 5  |-  ( ( x  =  X  /\  a  =  A )  ->  ( y R a  <-> 
y R A ) )
76abbidv 2590 . . . 4  |-  ( ( x  =  X  /\  a  =  A )  ->  { y  |  y R a }  =  { y  |  y R A } )
84, 7imaeq12d 5281 . . 3  |-  ( ( x  =  X  /\  a  =  A )  ->  ( `' x " { y  |  y R a } )  =  ( `' X " { y  |  y R A } ) )
98adantl 466 . 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 5548 . . . . 5  |-  ( x  =  X  ->  ( Fun  x  <->  Fun  X ) )
1312elabg 3214 . . . 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 3087 . . 3  |-  ( A  e.  W  ->  A  e.  _V )
1816, 17syl 16 . 2  |-  ( ph  ->  A  e.  _V )
19 cnvexg 6637 . . 3  |-  ( X  e.  V  ->  `' X  e.  _V )
20 imaexg 6628 . . 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 6331 1  |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  |  y R A } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2439   _Vcvv 3078   class class class wbr 4403   `'ccnv 4950   "cima 4954   Fun wfun 5523  (class class class)co 6203    |-> cmpt2 6205  ∘RV/𝑐corvc 27002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-orvc 27003
This theorem is referenced by:  orvcval2  27005  orvcval4  27007
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