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Theorem ovid 6193
Description: The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
ovid.1  |-  ( ( x  e.  R  /\  y  e.  S )  ->  E! z ph )
ovid.2  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
Assertion
Ref Expression
ovid  |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( ( x F y )  =  z  <->  ph ) )
Distinct variable groups:    x, y,
z    z, R    z, S
Allowed substitution hints:    ph( x, y, z)    R( x, y)    S( x, y)    F( x, y, z)

Proof of Theorem ovid
StepHypRef Expression
1 df-ov 6087 . . 3  |-  ( x F y )  =  ( F `  <. x ,  y >. )
21eqeq1i 2445 . 2  |-  ( ( x F y )  =  z  <->  ( F `  <. x ,  y
>. )  =  z
)
3 ovid.1 . . . . . 6  |-  ( ( x  e.  R  /\  y  e.  S )  ->  E! z ph )
43fnoprab 6176 . . . . 5  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S
)  /\  ph ) }  Fn  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) }
5 ovid.2 . . . . . 6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
65fneq1i 5542 . . . . 5  |-  ( F  Fn  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) } 
<->  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }  Fn  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) } )
74, 6mpbir 202 . . . 4  |-  F  Fn  {
<. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) }
8 opabid 4464 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  R  /\  y  e.  S ) }  <->  ( x  e.  R  /\  y  e.  S ) )
98biimpri 199 . . . 4  |-  ( ( x  e.  R  /\  y  e.  S )  -> 
<. x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  R  /\  y  e.  S ) } )
10 fnopfvb 5771 . . . 4  |-  ( ( F  Fn  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) }  /\  <. x ,  y
>.  e.  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) } )  ->  (
( F `  <. x ,  y >. )  =  z  <->  <. <. x ,  y
>. ,  z >.  e.  F ) )
117, 9, 10sylancr 646 . . 3  |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( ( F `  <. x ,  y >.
)  =  z  <->  <. <. x ,  y >. ,  z
>.  e.  F ) )
125eleq2i 2502 . . . . 5  |-  ( <. <. x ,  y >. ,  z >.  e.  F  <->  <. <. x ,  y >. ,  z >.  e.  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } )
13 oprabid 6108 . . . . 5  |-  ( <. <. x ,  y >. ,  z >.  e.  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }  <->  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) )
1412, 13bitri 242 . . . 4  |-  ( <. <. x ,  y >. ,  z >.  e.  F  <->  ( ( x  e.  R  /\  y  e.  S
)  /\  ph ) )
1514baib 873 . . 3  |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( <. <. x ,  y
>. ,  z >.  e.  F  <->  ph ) )
1611, 15bitrd 246 . 2  |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( ( F `  <. x ,  y >.
)  =  z  <->  ph ) )
172, 16syl5bb 250 1  |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( ( x F y )  =  z  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E!weu 2283   <.cop 3819   {copab 4268    Fn wfn 5452   ` cfv 5457  (class class class)co 6084   {coprab 6085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465  df-ov 6087  df-oprab 6088
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