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Theorem ovigg 5859
Description: The value of an operation class abstraction. Compare ovig 5860. The condition  ( x  e.  R  /\  y  e.  S ) is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ovigg.1  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
ovigg.4  |-  E* z ph
ovigg.5  |-  F  =  { <. <. x ,  y
>. ,  z >.  | 
ph }
Assertion
Ref Expression
ovigg  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ps  ->  ( A F B )  =  C ) )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    ps, x, y, z
Allowed substitution hints:    ph( x, y, z)    F( x, y, z)    V( x, y, z)    W( x, y, z)    X( x, y, z)

Proof of Theorem ovigg
StepHypRef Expression
1 ovigg.1 . . 3  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
21eloprabga 5826 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
3 df-ov 5745 . . . 4  |-  ( A F B )  =  ( F `  <. A ,  B >. )
4 ovigg.5 . . . . 5  |-  F  =  { <. <. x ,  y
>. ,  z >.  | 
ph }
54fveq1i 5390 . . . 4  |-  ( F `
 <. A ,  B >. )  =  ( {
<. <. x ,  y
>. ,  z >.  | 
ph } `  <. A ,  B >. )
63, 5eqtri 2138 . . 3  |-  ( A F B )  =  ( { <. <. x ,  y >. ,  z
>.  |  ph } `  <. A ,  B >. )
7 ovigg.4 . . . . 5  |-  E* z ph
87funoprab 5839 . . . 4  |-  Fun  { <. <. x ,  y
>. ,  z >.  | 
ph }
9 funopfv 5429 . . . 4  |-  ( Fun 
{ <. <. x ,  y
>. ,  z >.  | 
ph }  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  ->  ( { <. <. x ,  y
>. ,  z >.  | 
ph } `  <. A ,  B >. )  =  C ) )
108, 9ax-mp 5 . . 3  |-  ( <. <. A ,  B >. ,  C >.  e.  { <. <.
x ,  y >. ,  z >.  |  ph }  ->  ( { <. <.
x ,  y >. ,  z >.  |  ph } `  <. A ,  B >. )  =  C )
116, 10syl5eq 2162 . 2  |-  ( <. <. A ,  B >. ,  C >.  e.  { <. <.
x ,  y >. ,  z >.  |  ph }  ->  ( A F B )  =  C )
122, 11syl6bir 163 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ps  ->  ( A F B )  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   E*wmo 1978   <.cop 3500   Fun wfun 5087   ` cfv 5093  (class class class)co 5742   {coprab 5743
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-ov 5745  df-oprab 5746
This theorem is referenced by:  ovig  5860
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