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Theorem funoprabg 6292
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 4690 . . 3  |-  ( A. x A. y E* z ph  ->  E* z E. x E. y ( w  =  <. x ,  y >.  /\  ph ) )
21alrimiv 1686 . 2  |-  ( A. x A. y E* z ph  ->  A. w E* z E. x E. y ( w  =  <. x ,  y >.  /\  ph ) )
3 dfoprab2 6235 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
43funeqi 5539 . . 3  |-  ( Fun 
{ <. <. x ,  y
>. ,  z >.  | 
ph }  <->  Fun  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) } )
5 funopab 5552 . . 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 250 . 2  |-  ( A. w E* z E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  Fun  { <. <.
x ,  y >. ,  z >.  |  ph } )
72, 6sylib 196 1  |-  ( A. x A. y E* z ph  ->  Fun  { <. <. x ,  y >. ,  z
>.  |  ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1368    = wceq 1370   E.wex 1587   E*wmo 2261   <.cop 3984   {copab 4450   Fun wfun 5513   {coprab 6194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-fun 5521  df-oprab 6197
This theorem is referenced by:  funoprab  6293  fnoprabg  6294  oprabexd  6667
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