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Theorem orvcval2 23661
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 23660 . 2  |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  |  y R A } ) )
5 funfn 5285 . . . 4  |-  ( Fun 
X  <->  X  Fn  dom  X )
61, 5sylib 188 . . 3  |-  ( ph  ->  X  Fn  dom  X
)
7 fncnvima2 5649 . . 3  |-  ( X  Fn  dom  X  -> 
( `' X " { y  |  y R A } )  =  { z  e. 
dom  X  |  ( X `  z )  e.  { y  |  y R A } }
)
86, 7syl 15 . 2  |-  ( ph  ->  ( `' X " { y  |  y R A } )  =  { z  e. 
dom  X  |  ( X `  z )  e.  { y  |  y R A } }
)
9 fvex 5541 . . . . . 6  |-  ( X `
 z )  e. 
_V
10 breq1 4028 . . . . . 6  |-  ( y  =  ( X `  z )  ->  (
y R A  <->  ( X `  z ) R A ) )
119, 10elab 2916 . . . . 5  |-  ( ( X `  z )  e.  { y  |  y R A }  <->  ( X `  z ) R A )
1211a1i 10 . . . 4  |-  ( z  e.  dom  X  -> 
( ( X `  z )  e.  {
y  |  y R A }  <->  ( X `  z ) R A ) )
1312rabbiia 2780 . . 3  |-  { z  e.  dom  X  | 
( X `  z
)  e.  { y  |  y R A } }  =  {
z  e.  dom  X  |  ( X `  z ) R A }
1413a1i 10 . 2  |-  ( ph  ->  { z  e.  dom  X  |  ( X `  z )  e.  {
y  |  y R A } }  =  { z  e.  dom  X  |  ( X `  z ) R A } )
154, 8, 143eqtrd 2321 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 176    = wceq 1625    e. wcel 1686   {cab 2271   {crab 2549   class class class wbr 4025   `'ccnv 4690   dom cdm 4691   "cima 4694   Fun wfun 5251    Fn wfn 5252   ` cfv 5257  (class class class)co 5860  ∘RV/𝑐corvc 23658
This theorem is referenced by:  elorvc  23662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-orvc 23659
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