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Theorem ovigg 5747
Description: The value of an operation class abstraction. Compare ovig 5748. 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 5717 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
3 df-ov 5637 . . . 4  |-  ( A F B )  =  ( F `  <. A ,  B >. )
4 ovigg.5 . . . . 5  |-  F  =  { <. <. x ,  y
>. ,  z >.  | 
ph }
54fveq1i 5290 . . . 4  |-  ( F `
 <. A ,  B >. )  =  ( {
<. <. x ,  y
>. ,  z >.  | 
ph } `  <. A ,  B >. )
63, 5eqtri 2108 . . 3  |-  ( A F B )  =  ( { <. <. x ,  y >. ,  z
>.  |  ph } `  <. A ,  B >. )
7 ovigg.4 . . . . 5  |-  E* z ph
87funoprab 5727 . . . 4  |-  Fun  { <. <. x ,  y
>. ,  z >.  | 
ph }
9 funopfv 5328 . . . 4  |-  ( Fun 
{ <. <. x ,  y
>. ,  z >.  | 
ph }  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  ->  ( { <. <. x ,  y
>. ,  z >.  | 
ph } `  <. A ,  B >. )  =  C ) )
108, 9ax-mp 7 . . 3  |-  ( <. <. A ,  B >. ,  C >.  e.  { <. <.
x ,  y >. ,  z >.  |  ph }  ->  ( { <. <.
x ,  y >. ,  z >.  |  ph } `  <. A ,  B >. )  =  C )
116, 10syl5eq 2132 . 2  |-  ( <. <. A ,  B >. ,  C >.  e.  { <. <.
x ,  y >. ,  z >.  |  ph }  ->  ( A F B )  =  C )
122, 11syl6bir 162 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 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E*wmo 1949   <.cop 3444   Fun wfun 4996   ` cfv 5002  (class class class)co 5634   {coprab 5635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-ov 5637  df-oprab 5638
This theorem is referenced by:  ovig  5748
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