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Theorem ovig 5650
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 912 . 2  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( A  e.  R  /\  B  e.  S
) )
2 eleq1 2116 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  R  <->  A  e.  R ) )
3 eleq1 2116 . . . . . 6  |-  ( y  =  B  ->  (
y  e.  S  <->  B  e.  S ) )
42, 3bi2anan9 548 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  e.  R  /\  y  e.  S )  <->  ( A  e.  R  /\  B  e.  S ) ) )
543adant3 935 . . . 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 450 . . 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 1991 . . . 4  |-  ( E* z ( ( x  e.  R  /\  y  e.  S )  /\  ph ) 
<->  ( ( x  e.  R  /\  y  e.  S )  ->  E* z ph ) )
108, 9mpbir 138 . . 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 5649 . 2  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( ( ( A  e.  R  /\  B  e.  S )  /\  ps )  ->  ( A F B )  =  C ) )
131, 12mpand 413 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 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   E*wmo 1917  (class class class)co 5540   {coprab 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544
This theorem is referenced by:  th3q  6242  addnnnq0  6605  mulnnnq0  6606  addsrpr  6888  mulsrpr  6889
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