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Theorem ovigg 5984
Description: The value of an operation class abstraction. Compare ovig 5985. 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 5950 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
3 df-ov 5877 . . . 4  |-  ( A F B )  =  ( F `  <. A ,  B >. )
4 ovigg.5 . . . . 5  |-  F  =  { <. <. x ,  y
>. ,  z >.  | 
ph }
54fveq1i 5542 . . . 4  |-  ( F `
 <. A ,  B >. )  =  ( {
<. <. x ,  y
>. ,  z >.  | 
ph } `  <. A ,  B >. )
63, 5eqtri 2316 . . 3  |-  ( A F B )  =  ( { <. <. x ,  y >. ,  z
>.  |  ph } `  <. A ,  B >. )
7 ovigg.4 . . . . 5  |-  E* z ph
87funoprab 5960 . . . 4  |-  Fun  { <. <. x ,  y
>. ,  z >.  | 
ph }
9 funopfv 5578 . . . 4  |-  ( Fun 
{ <. <. x ,  y
>. ,  z >.  | 
ph }  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  ->  ( { <. <. x ,  y
>. ,  z >.  | 
ph } `  <. A ,  B >. )  =  C ) )
108, 9ax-mp 8 . . 3  |-  ( <. <. A ,  B >. ,  C >.  e.  { <. <.
x ,  y >. ,  z >.  |  ph }  ->  ( { <. <.
x ,  y >. ,  z >.  |  ph } `  <. A ,  B >. )  =  C )
116, 10syl5eq 2340 . 2  |-  ( <. <. A ,  B >. ,  C >.  e.  { <. <.
x ,  y >. ,  z >.  |  ph }  ->  ( A F B )  =  C )
122, 11syl6bir 220 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 176    /\ w3a 934    = wceq 1632    e. wcel 1696   E*wmo 2157   <.cop 3656   Fun wfun 5265   ` cfv 5271  (class class class)co 5874   {coprab 5875
This theorem is referenced by:  ovig  5985  cmp2morp  26061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878
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