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

Theorem ovidi 5982
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 2214 . . . 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 5981 . 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 1632    e. wcel 1696   E*wmo 2157  (class class class)co 5874   {coprab 5875
This theorem is referenced by:  ovmpt4g  5986  ov3  6000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878
  Copyright terms: Public domain W3C validator