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Theorem ovidi 6184
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 2338 . . . 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 6183 . 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 1652    e. wcel 1725   E*wmo 2281  (class class class)co 6073   {coprab 6074
This theorem is referenced by:  ovmpt4g  6188  ov3  6202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077
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