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Theorem ov 6009
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 5903 . . . . 5  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 ov.6 . . . . . 6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
32fveq1i 5564 . . . . 5  |-  ( F `
 <. A ,  B >. )  =  ( {
<. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )
41, 3eqtri 2336 . . . 4  |-  ( A F B )  =  ( { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )
54eqeq1i 2323 . . 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 5989 . . . . 5  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S
)  /\  ph ) }  Fn  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) }
8 eleq1 2376 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  R  <->  A  e.  R ) )
98anbi1d 685 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  R  /\  y  e.  S
)  <->  ( A  e.  R  /\  y  e.  S ) ) )
10 eleq1 2376 . . . . . . . 8  |-  ( y  =  B  ->  (
y  e.  S  <->  B  e.  S ) )
1110anbi2d 684 . . . . . . 7  |-  ( y  =  B  ->  (
( A  e.  R  /\  y  e.  S
)  <->  ( A  e.  R  /\  B  e.  S ) ) )
129, 11opelopabg 4320 . . . . . 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 233 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  R  /\  y  e.  S ) } )
14 fnopfvb 5602 . . . . 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 644 . . . 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 691 . . . . . 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 691 . . . . . 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 684 . . . . . 6  |-  ( z  =  C  ->  (
( ( A  e.  R  /\  B  e.  S )  /\  ch ) 
<->  ( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2318, 20, 22eloprabg 5977 . . . . 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 1266 . . . 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 244 . . 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 248 . 2  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A F B )  =  C  <-> 
( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2726bianabs 850 1  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A F B )  =  C  <->  th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   E!weu 2176   _Vcvv 2822   <.cop 3677   {copab 4113    Fn wfn 5287   ` cfv 5292  (class class class)co 5900   {coprab 5901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fn 5295  df-fv 5300  df-ov 5903  df-oprab 5904
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