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Theorem funoprabg 6132
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 4418 . . 3  |-  ( A. x A. y E* z ph  ->  E* z E. x E. y ( w  =  <. x ,  y >.  /\  ph ) )
21alrimiv 1638 . 2  |-  ( A. x A. y E* z ph  ->  A. w E* z E. x E. y ( w  =  <. x ,  y >.  /\  ph ) )
3 dfoprab2 6084 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
43funeqi 5437 . . 3  |-  ( Fun 
{ <. <. x ,  y
>. ,  z >.  | 
ph }  <->  Fun  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) } )
5 funopab 5449 . . 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 242 . 2  |-  ( A. w E* z E. x E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  Fun  { <. <.
x ,  y >. ,  z >.  |  ph } )
72, 6sylib 189 1  |-  ( A. x A. y E* z ph  ->  Fun  { <. <. x ,  y >. ,  z
>.  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649   E*wmo 2259   <.cop 3781   {copab 4229   Fun wfun 5411   {coprab 6045
This theorem is referenced by:  funoprab  6133  fnoprabg  6134  oprabexd  6149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-fun 5419  df-oprab 6048
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