MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovidi Unicode version

Theorem ovidi 5966
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 2201 . . . 4  |-  ( E* z ( ( x  e.  R  /\  y  e.  S )  /\  ph ) 
<->  ( ( x  e.  R  /\  y  e.  S )  ->  E* z ph ) )
31, 2mpbir 200 . . 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 5965 . 2  |-  ( ( ( x  e.  R  /\  y  e.  S
)  /\  ph )  -> 
( x F y )  =  z )
65ex 423 1  |-  ( ( x  e.  R  /\  y  e.  S )  ->  ( ph  ->  (
x F y )  =  z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E*wmo 2144  (class class class)co 5858   {coprab 5859
This theorem is referenced by:  ovmpt4g  5970  ov3  5984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862
  Copyright terms: Public domain W3C validator