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Theorem ovig 5748
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 940 . 2  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( A  e.  R  /\  B  e.  S
) )
2 eleq1 2150 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  R  <->  A  e.  R ) )
3 eleq1 2150 . . . . . 6  |-  ( y  =  B  ->  (
y  e.  S  <->  B  e.  S ) )
42, 3bi2anan9 573 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  e.  R  /\  y  e.  S )  <->  ( A  e.  R  /\  B  e.  S ) ) )
543adant3 963 . . . 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 457 . . 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 2023 . . . 4  |-  ( E* z ( ( x  e.  R  /\  y  e.  S )  /\  ph ) 
<->  ( ( x  e.  R  /\  y  e.  S )  ->  E* z ph ) )
108, 9mpbir 144 . . 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 5747 . 2  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( ( ( A  e.  R  /\  B  e.  S )  /\  ps )  ->  ( A F B )  =  C ) )
131, 12mpand 420 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E*wmo 1949  (class class class)co 5634   {coprab 5635
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-ov 5637  df-oprab 5638
This theorem is referenced by:  th3q  6377  addnnnq0  6987  mulnnnq0  6988  addsrpr  7270  mulsrpr  7271
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