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Theorem ov 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
ov.1  |-  C  e. 
_V
ov.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ov.3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
ov.4  |-  ( z  =  C  ->  ( ch 
<->  th ) )
ov.5  |-  ( ( x  e.  R  /\  y  e.  S )  ->  E! z ph )
ov.6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
Assertion
Ref Expression
ov  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A F B )  =  C  <->  th ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    x, R, y, z    x, S, y, z    th, x, y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)    ch( x, y, z)    F( x, y, z)

Proof of Theorem ov
StepHypRef Expression
1 df-ov 6084 . . . . 5  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 ov.6 . . . . . 6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
32fveq1i 5729 . . . . 5  |-  ( F `
 <. A ,  B >. )  =  ( {
<. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )
41, 3eqtri 2456 . . . 4  |-  ( A F B )  =  ( { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )
54eqeq1i 2443 . . 3  |-  ( ( A F B )  =  C  <->  ( { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )  =  C )
6 ov.5 . . . . . 6  |-  ( ( x  e.  R  /\  y  e.  S )  ->  E! z ph )
76fnoprab 6173 . . . . 5  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S
)  /\  ph ) }  Fn  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) }
8 eleq1 2496 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  R  <->  A  e.  R ) )
98anbi1d 686 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  R  /\  y  e.  S
)  <->  ( A  e.  R  /\  y  e.  S ) ) )
10 eleq1 2496 . . . . . . . 8  |-  ( y  =  B  ->  (
y  e.  S  <->  B  e.  S ) )
1110anbi2d 685 . . . . . . 7  |-  ( y  =  B  ->  (
( A  e.  R  /\  y  e.  S
)  <->  ( A  e.  R  /\  B  e.  S ) ) )
129, 11opelopabg 4473 . . . . . 6  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) } 
<->  ( A  e.  R  /\  B  e.  S
) ) )
1312ibir 234 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  R  /\  y  e.  S ) } )
14 fnopfvb 5768 . . . . 5  |-  ( ( { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }  Fn  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) }  /\  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) } )  ->  (
( { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )  =  C  <->  <. <. A ,  B >. ,  C >.  e. 
{ <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } ) )
157, 13, 14sylancr 645 . . . 4  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S
)  /\  ph ) } `
 <. A ,  B >. )  =  C  <->  <. <. A ,  B >. ,  C >.  e. 
{ <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } ) )
16 ov.1 . . . . 5  |-  C  e. 
_V
17 ov.2 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
189, 17anbi12d 692 . . . . . 6  |-  ( x  =  A  ->  (
( ( x  e.  R  /\  y  e.  S )  /\  ph ) 
<->  ( ( A  e.  R  /\  y  e.  S )  /\  ps ) ) )
19 ov.3 . . . . . . 7  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
2011, 19anbi12d 692 . . . . . 6  |-  ( y  =  B  ->  (
( ( A  e.  R  /\  y  e.  S )  /\  ps ) 
<->  ( ( A  e.  R  /\  B  e.  S )  /\  ch ) ) )
21 ov.4 . . . . . . 7  |-  ( z  =  C  ->  ( ch 
<->  th ) )
2221anbi2d 685 . . . . . 6  |-  ( z  =  C  ->  (
( ( A  e.  R  /\  B  e.  S )  /\  ch ) 
<->  ( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2318, 20, 22eloprabg 6161 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  _V )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }  <->  ( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2416, 23mp3an3 1268 . . . 4  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }  <->  ( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2515, 24bitrd 245 . . 3  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S
)  /\  ph ) } `
 <. A ,  B >. )  =  C  <->  ( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
265, 25syl5bb 249 . 2  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A F B )  =  C  <-> 
( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2726bianabs 851 1  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A F B )  =  C  <->  th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E!weu 2281   _Vcvv 2956   <.cop 3817   {copab 4265    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   {coprab 6082
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-ov 6084  df-oprab 6085
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