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Theorem ov 5940
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 5827 . . . . 5  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 ov.6 . . . . . 6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
32fveq1i 5469 . . . . 5  |-  ( F `
 <. A ,  B >. )  =  ( {
<. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )
41, 3eqtri 2178 . . . 4  |-  ( A F B )  =  ( { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )
54eqeq1i 2165 . . 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 5924 . . . . 5  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S
)  /\  ph ) }  Fn  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) }
8 eleq1 2220 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  R  <->  A  e.  R ) )
98anbi1d 461 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  R  /\  y  e.  S
)  <->  ( A  e.  R  /\  y  e.  S ) ) )
10 eleq1 2220 . . . . . . . 8  |-  ( y  =  B  ->  (
y  e.  S  <->  B  e.  S ) )
1110anbi2d 460 . . . . . . 7  |-  ( y  =  B  ->  (
( A  e.  R  /\  y  e.  S
)  <->  ( A  e.  R  /\  B  e.  S ) ) )
129, 11opelopabg 4228 . . . . . 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 176 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  R  /\  y  e.  S ) } )
14 fnopfvb 5510 . . . . 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 411 . . . 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 465 . . . . . 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 465 . . . . . 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 460 . . . . . 6  |-  ( z  =  C  ->  (
( ( A  e.  R  /\  B  e.  S )  /\  ch ) 
<->  ( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2318, 20, 22eloprabg 5909 . . . . 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 1308 . . . 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 187 . . 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 191 . 2  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A F B )  =  C  <-> 
( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2726bianabs 601 1  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A F B )  =  C  <->  th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335   E!weu 2006    e. wcel 2128   _Vcvv 2712   <.cop 3563   {copab 4024    Fn wfn 5165   ` cfv 5170  (class class class)co 5824   {coprab 5825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fn 5173  df-fv 5178  df-ov 5827  df-oprab 5828
This theorem is referenced by: (None)
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