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Theorem ovig 6397
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 991 . 2  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( A  e.  R  /\  B  e.  S
) )
2 eleq1 2526 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  R  <->  A  e.  R ) )
3 eleq1 2526 . . . . . 6  |-  ( y  =  B  ->  (
y  e.  S  <->  B  e.  S ) )
42, 3bi2anan9 871 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  e.  R  /\  y  e.  S )  <->  ( A  e.  R  /\  B  e.  S ) ) )
543adant3 1014 . . . 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 708 . . 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 2349 . . . 4  |-  ( E* z ( ( x  e.  R  /\  y  e.  S )  /\  ph ) 
<->  ( ( x  e.  R  /\  y  e.  S )  ->  E* z ph ) )
108, 9mpbir 209 . . 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 6396 . 2  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( ( ( A  e.  R  /\  B  e.  S )  /\  ps )  ->  ( A F B )  =  C ) )
131, 12mpand 673 1  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( ps  ->  ( A F B )  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E*wmo 2285  (class class class)co 6270   {coprab 6271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274
This theorem is referenced by:  addsrpr  9441  mulsrpr  9442
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