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Theorem ovigg 6322
Description: The value of an operation class abstraction. Compare ovig 6323. 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 6288 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
3 df-ov 6199 . . . 4  |-  ( A F B )  =  ( F `  <. A ,  B >. )
4 ovigg.5 . . . . 5  |-  F  =  { <. <. x ,  y
>. ,  z >.  | 
ph }
54fveq1i 5775 . . . 4  |-  ( F `
 <. A ,  B >. )  =  ( {
<. <. x ,  y
>. ,  z >.  | 
ph } `  <. A ,  B >. )
63, 5eqtri 2411 . . 3  |-  ( A F B )  =  ( { <. <. x ,  y >. ,  z
>.  |  ph } `  <. A ,  B >. )
7 ovigg.4 . . . . 5  |-  E* z ph
87funoprab 6301 . . . 4  |-  Fun  { <. <. x ,  y
>. ,  z >.  | 
ph }
9 funopfv 5813 . . . 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 2435 . 2  |-  ( <. <. A ,  B >. ,  C >.  e.  { <. <.
x ,  y >. ,  z >.  |  ph }  ->  ( A F B )  =  C )
122, 11syl6bir 229 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ps  ->  ( A F B )  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1399    e. wcel 1826   E*wmo 2219   <.cop 3950   Fun wfun 5490   ` cfv 5496  (class class class)co 6196   {coprab 6197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-oprab 6200
This theorem is referenced by:  ovig  6323  joinval  15752  meetval  15766
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