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Theorem orvcval2 24496
Description: Another way to express the value of the preimage mapping operator (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
orvcval2  |-  ( ph  ->  ( XRV/𝑐 R A )  =  {
z  e.  dom  X  |  ( X `  z ) R A } )
Distinct variable groups:    z, A    z, R    z, X
Allowed substitution hints:    ph( z)    V( z)    W( z)

Proof of Theorem orvcval2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 orvcval.1 . . 3  |-  ( ph  ->  Fun  X )
2 orvcval.2 . . 3  |-  ( ph  ->  X  e.  V )
3 orvcval.3 . . 3  |-  ( ph  ->  A  e.  W )
41, 2, 3orvcval 24495 . 2  |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  |  y R A } ) )
5 funfn 5423 . . . 4  |-  ( Fun 
X  <->  X  Fn  dom  X )
61, 5sylib 189 . . 3  |-  ( ph  ->  X  Fn  dom  X
)
7 fncnvima2 5792 . . 3  |-  ( X  Fn  dom  X  -> 
( `' X " { y  |  y R A } )  =  { z  e. 
dom  X  |  ( X `  z )  e.  { y  |  y R A } }
)
86, 7syl 16 . 2  |-  ( ph  ->  ( `' X " { y  |  y R A } )  =  { z  e. 
dom  X  |  ( X `  z )  e.  { y  |  y R A } }
)
9 fvex 5683 . . . . . 6  |-  ( X `
 z )  e. 
_V
10 breq1 4157 . . . . . 6  |-  ( y  =  ( X `  z )  ->  (
y R A  <->  ( X `  z ) R A ) )
119, 10elab 3026 . . . . 5  |-  ( ( X `  z )  e.  { y  |  y R A }  <->  ( X `  z ) R A )
1211a1i 11 . . . 4  |-  ( z  e.  dom  X  -> 
( ( X `  z )  e.  {
y  |  y R A }  <->  ( X `  z ) R A ) )
1312rabbiia 2890 . . 3  |-  { z  e.  dom  X  | 
( X `  z
)  e.  { y  |  y R A } }  =  {
z  e.  dom  X  |  ( X `  z ) R A }
1413a1i 11 . 2  |-  ( ph  ->  { z  e.  dom  X  |  ( X `  z )  e.  {
y  |  y R A } }  =  { z  e.  dom  X  |  ( X `  z ) R A } )
154, 8, 143eqtrd 2424 1  |-  ( ph  ->  ( XRV/𝑐 R A )  =  {
z  e.  dom  X  |  ( X `  z ) R A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   {cab 2374   {crab 2654   class class class wbr 4154   `'ccnv 4818   dom cdm 4819   "cima 4822   Fun wfun 5389    Fn wfn 5390   ` cfv 5395  (class class class)co 6021  ∘RV/𝑐corvc 24493
This theorem is referenced by:  elorvc  24497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-orvc 24494
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