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Theorem fnovpop 25038
Description: Representation of an operation class abstraction in terms of its values. (A version of fnov 5952 adapted to partial operations.) (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
fnovpop  |-  ( Rel 
R  ->  ( F  Fn  R  <->  F  =  { <. <. x ,  y
>. ,  z >.  |  ( <. x ,  y
>.  e.  R  /\  z  =  ( x F y ) ) } ) )
Distinct variable groups:    x, F, y, z    x, R, y, z

Proof of Theorem fnovpop
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dffn5 5568 . 2  |-  ( F  Fn  R  <->  F  =  ( w  e.  R  |->  ( F `  w
) ) )
2 df-mpt 4079 . . . 4  |-  ( w  e.  R  |->  ( F `
 w ) )  =  { <. w ,  z >.  |  ( w  e.  R  /\  z  =  ( F `  w ) ) }
3 fveq2 5525 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( F `  <. x ,  y >. )
)
4 df-ov 5861 . . . . . . 7  |-  ( x F y )  =  ( F `  <. x ,  y >. )
53, 4syl6eqr 2333 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( x F y ) )
65eqeq2d 2294 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( F `  w
)  <->  z  =  ( x F y ) ) )
76dfoprab4pop 25037 . . . 4  |-  ( Rel 
R  ->  { <. w ,  z >.  |  ( w  e.  R  /\  z  =  ( F `  w ) ) }  =  { <. <. x ,  y >. ,  z
>.  |  ( <. x ,  y >.  e.  R  /\  z  =  (
x F y ) ) } )
82, 7syl5eq 2327 . . 3  |-  ( Rel 
R  ->  ( w  e.  R  |->  ( F `
 w ) )  =  { <. <. x ,  y >. ,  z
>.  |  ( <. x ,  y >.  e.  R  /\  z  =  (
x F y ) ) } )
98eqeq2d 2294 . 2  |-  ( Rel 
R  ->  ( F  =  ( w  e.  R  |->  ( F `  w ) )  <->  F  =  { <. <. x ,  y
>. ,  z >.  |  ( <. x ,  y
>.  e.  R  /\  z  =  ( x F y ) ) } ) )
101, 9syl5bb 248 1  |-  ( Rel 
R  ->  ( F  Fn  R  <->  F  =  { <. <. x ,  y
>. ,  z >.  |  ( <. x ,  y
>.  e.  R  /\  z  =  ( x F y ) ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   {copab 4076    e. cmpt 4077   Rel wrel 4694    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   {coprab 5859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861  df-oprab 5862
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