MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovigg Unicode version

Theorem ovigg 6135
Description: The value of an operation class abstraction. Compare ovig 6136. 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 6101 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
3 df-ov 6025 . . . 4  |-  ( A F B )  =  ( F `  <. A ,  B >. )
4 ovigg.5 . . . . 5  |-  F  =  { <. <. x ,  y
>. ,  z >.  | 
ph }
54fveq1i 5671 . . . 4  |-  ( F `
 <. A ,  B >. )  =  ( {
<. <. x ,  y
>. ,  z >.  | 
ph } `  <. A ,  B >. )
63, 5eqtri 2409 . . 3  |-  ( A F B )  =  ( { <. <. x ,  y >. ,  z
>.  |  ph } `  <. A ,  B >. )
7 ovigg.4 . . . . 5  |-  E* z ph
87funoprab 6111 . . . 4  |-  Fun  { <. <. x ,  y
>. ,  z >.  | 
ph }
9 funopfv 5707 . . . 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 2433 . 2  |-  ( <. <. A ,  B >. ,  C >.  e.  { <. <.
x ,  y >. ,  z >.  |  ph }  ->  ( A F B )  =  C )
122, 11syl6bir 221 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 177    /\ w3a 936    = wceq 1649    e. wcel 1717   E*wmo 2241   <.cop 3762   Fun wfun 5390   ` cfv 5396  (class class class)co 6022   {coprab 6023
This theorem is referenced by:  ovig  6136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fv 5404  df-ov 6025  df-oprab 6026
  Copyright terms: Public domain W3C validator