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Theorem orvcval2 24708
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 24707 . 2  |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  |  y R A } ) )
5 funfn 5474 . . . 4  |-  ( Fun 
X  <->  X  Fn  dom  X )
61, 5sylib 189 . . 3  |-  ( ph  ->  X  Fn  dom  X
)
7 fncnvima2 5844 . . 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 5734 . . . . . 6  |-  ( X `
 z )  e. 
_V
10 breq1 4207 . . . . . 6  |-  ( y  =  ( X `  z )  ->  (
y R A  <->  ( X `  z ) R A ) )
119, 10elab 3074 . . . . 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 2938 . . 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 2471 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 1652    e. wcel 1725   {cab 2421   {crab 2701   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   "cima 4873   Fun wfun 5440    Fn wfn 5441   ` cfv 5446  (class class class)co 6073  ∘RV/𝑐corvc 24705
This theorem is referenced by:  elorvc  24709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-orvc 24706
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