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Theorem funoprabg 5943
Description: "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.)
Assertion
Ref Expression
funoprabg  |-  ( A. x A. y E* z ph  ->  Fun  { <. <. x ,  y >. ,  z
>.  |  ph } )
Distinct variable group:    x, y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem funoprabg
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 mosubopt 4264 . . 3  |-  ( A. x A. y E* z ph  ->  E* z E. x E. y ( w  =  <. x ,  y >.  /\  ph ) )
21alrimiv 1617 . 2  |-  ( A. x A. y E* z ph  ->  A. w E* z E. x E. y ( w  =  <. x ,  y >.  /\  ph ) )
3 dfoprab2 5895 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
43funeqi 5275 . . 3  |-  ( Fun 
{ <. <. x ,  y
>. ,  z >.  | 
ph }  <->  Fun  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) } )
5 funopab 5287 . . 3  |-  ( Fun 
{ <. w ,  z
>.  |  E. x E. y ( w  = 
<. x ,  y >.  /\  ph ) }  <->  A. w E* z E. x E. y ( w  = 
<. x ,  y >.  /\  ph ) )
64, 5bitr2i 241 . 2  |-  ( A. w E* z E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  Fun  { <. <.
x ,  y >. ,  z >.  |  ph } )
72, 6sylib 188 1  |-  ( A. x A. y E* z ph  ->  Fun  { <. <. x ,  y >. ,  z
>.  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623   E*wmo 2144   <.cop 3643   {copab 4076   Fun wfun 5249   {coprab 5859
This theorem is referenced by:  funoprab  5944  fnoprabg  5945  oprabexd  5960
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-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-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-fun 5257  df-oprab 5862
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