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Theorem ovig 5858
Description: The value of an operation class abstraction (weak version). (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ovig.1  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
ovig.2  |-  ( ( x  e.  R  /\  y  e.  S )  ->  E* z ph )
ovig.3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
Assertion
Ref Expression
ovig  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( ps  ->  ( A F B )  =  C ) )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    x, R, y, z    x, S, y, z    ps, x, y, z
Allowed substitution hints:    ph( x, y, z)    D( x, y, z)    F( x, y, z)

Proof of Theorem ovig
StepHypRef Expression
1 3simpa 961 . 2  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( A  e.  R  /\  B  e.  S
) )
2 eleq1 2178 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  R  <->  A  e.  R ) )
3 eleq1 2178 . . . . . 6  |-  ( y  =  B  ->  (
y  e.  S  <->  B  e.  S ) )
42, 3bi2anan9 578 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  e.  R  /\  y  e.  S )  <->  ( A  e.  R  /\  B  e.  S ) ) )
543adant3 984 . . . 4  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( x  e.  R  /\  y  e.  S )  <->  ( A  e.  R  /\  B  e.  S ) ) )
6 ovig.1 . . . 4  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
75, 6anbi12d 462 . . 3  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( ( x  e.  R  /\  y  e.  S )  /\  ph ) 
<->  ( ( A  e.  R  /\  B  e.  S )  /\  ps ) ) )
8 ovig.2 . . . 4  |-  ( ( x  e.  R  /\  y  e.  S )  ->  E* z ph )
9 moanimv 2050 . . . 4  |-  ( E* z ( ( x  e.  R  /\  y  e.  S )  /\  ph ) 
<->  ( ( x  e.  R  /\  y  e.  S )  ->  E* z ph ) )
108, 9mpbir 145 . . 3  |-  E* z
( ( x  e.  R  /\  y  e.  S )  /\  ph )
11 ovig.3 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
127, 10, 11ovigg 5857 . 2  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( ( ( A  e.  R  /\  B  e.  S )  /\  ps )  ->  ( A F B )  =  C ) )
131, 12mpand 423 1  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( ps  ->  ( A F B )  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   E*wmo 1976  (class class class)co 5740   {coprab 5741
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-ov 5743  df-oprab 5744
This theorem is referenced by:  th3q  6500  addnnnq0  7221  mulnnnq0  7222  addsrpr  7517  mulsrpr  7518
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