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Theorem ovig 6231
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 985 . 2  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( A  e.  R  /\  B  e.  S
) )
2 eleq1 2503 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  R  <->  A  e.  R ) )
3 eleq1 2503 . . . . . 6  |-  ( y  =  B  ->  (
y  e.  S  <->  B  e.  S ) )
42, 3bi2anan9 868 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  e.  R  /\  y  e.  S )  <->  ( A  e.  R  /\  B  e.  S ) ) )
543adant3 1008 . . . 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 710 . . 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 2334 . . . 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 6230 . 2  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  D )  ->  ( ( ( A  e.  R  /\  B  e.  S )  /\  ps )  ->  ( A F B )  =  C ) )
131, 12mpand 675 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E*wmo 2254  (class class class)co 6110   {coprab 6111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-iota 5400  df-fun 5439  df-fv 5445  df-ov 6113  df-oprab 6114
This theorem is referenced by:  th3q  7228
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