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Theorem ovidi 6131
Description: The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovidi.2  |-  ( ( x  e.  R  /\  y  e.  S )  ->  E* z ph )
ovidi.3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
Assertion
Ref Expression
ovidi  |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( ph  ->  (
x F y )  =  z ) )
Distinct variable groups:    x, y,
z    z, R    z, S
Allowed substitution hints:    ph( x, y, z)    R( x, y)    S( x, y)    F( x, y, z)

Proof of Theorem ovidi
StepHypRef Expression
1 ovidi.2 . . . 4  |-  ( ( x  e.  R  /\  y  e.  S )  ->  E* z ph )
2 moanimv 2296 . . . 4  |-  ( E* z ( ( x  e.  R  /\  y  e.  S )  /\  ph ) 
<->  ( ( x  e.  R  /\  y  e.  S )  ->  E* z ph ) )
31, 2mpbir 201 . . 3  |-  E* z
( ( x  e.  R  /\  y  e.  S )  /\  ph )
4 ovidi.3 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
53, 4ovidig 6130 . 2  |-  ( ( ( x  e.  R  /\  y  e.  S
)  /\  ph )  -> 
( x F y )  =  z )
65ex 424 1  |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( ph  ->  (
x F y )  =  z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   E*wmo 2239  (class class class)co 6020   {coprab 6021
This theorem is referenced by:  ovmpt4g  6135  ov3  6149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-oprab 6024
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