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Theorem ovid 6051
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 5948 . . 3  |-  ( x F y )  =  ( F `  <. x ,  y >. )
21eqeq1i 2365 . 2  |-  ( ( x F y )  =  z  <->  ( F `  <. x ,  y
>. )  =  z
)
3 ovid.1 . . . . . 6  |-  ( ( x  e.  R  /\  y  e.  S )  ->  E! z ph )
43fnoprab 6034 . . . . 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 5420 . . . . 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 200 . . . 4  |-  F  Fn  {
<. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) }
8 opabid 4353 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  R  /\  y  e.  S ) }  <->  ( x  e.  R  /\  y  e.  S ) )
98biimpri 197 . . . 4  |-  ( ( x  e.  R  /\  y  e.  S )  -> 
<. x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  R  /\  y  e.  S ) } )
10 fnopfvb 5647 . . . 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 644 . . 3  |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( ( F `  <. x ,  y >.
)  =  z  <->  <. <. x ,  y >. ,  z
>.  e.  F ) )
125eleq2i 2422 . . . . 5  |-  ( <. <. x ,  y >. ,  z >.  e.  F  <->  <. <. x ,  y >. ,  z >.  e.  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } )
13 oprabid 5969 . . . . 5  |-  ( <. <. x ,  y >. ,  z >.  e.  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }  <->  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) )
1412, 13bitri 240 . . . 4  |-  ( <. <. x ,  y >. ,  z >.  e.  F  <->  ( ( x  e.  R  /\  y  e.  S
)  /\  ph ) )
1514baib 871 . . 3  |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( <. <. x ,  y
>. ,  z >.  e.  F  <->  ph ) )
1611, 15bitrd 244 . 2  |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( ( F `  <. x ,  y >.
)  =  z  <->  ph ) )
172, 16syl5bb 248 1  |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( ( x F y )  =  z  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   E!weu 2209   <.cop 3719   {copab 4157    Fn wfn 5332   ` cfv 5337  (class class class)co 5945   {coprab 5946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fn 5340  df-fv 5345  df-ov 5948  df-oprab 5949
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