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Theorem funoprabg 6069
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 4367 . . 3  |-  ( A. x A. y E* z ph  ->  E* z E. x E. y ( w  =  <. x ,  y >.  /\  ph ) )
21alrimiv 1636 . 2  |-  ( A. x A. y E* z ph  ->  A. w E* z E. x E. y ( w  =  <. x ,  y >.  /\  ph ) )
3 dfoprab2 6021 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) }
43funeqi 5378 . . 3  |-  ( Fun 
{ <. <. x ,  y
>. ,  z >.  | 
ph }  <->  Fun  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ph ) } )
5 funopab 5390 . . 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 1545   E.wex 1546    = wceq 1647   E*wmo 2218   <.cop 3732   {copab 4178   Fun wfun 5352   {coprab 5982
This theorem is referenced by:  funoprab  6070  fnoprabg  6071  oprabexd  6086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-fun 5360  df-oprab 5985
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