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Theorem dffun3 5457
Description: Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
dffun3  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E. z A. y ( x A y  ->  y  =  z ) ) )
Distinct variable group:    x, y, z, A

Proof of Theorem dffun3
StepHypRef Expression
1 dffun2 5456 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x A. y A. z ( ( x A y  /\  x A z )  -> 
y  =  z ) ) )
2 breq2 4208 . . . . . 6  |-  ( y  =  z  ->  (
x A y  <->  x A
z ) )
32mo4 2313 . . . . 5  |-  ( E* y  x A y  <->  A. y A. z ( ( x A y  /\  x A z )  ->  y  =  z ) )
4 nfv 1629 . . . . . 6  |-  F/ z  x A y
54mo2 2309 . . . . 5  |-  ( E* y  x A y  <->  E. z A. y ( x A y  -> 
y  =  z ) )
63, 5bitr3i 243 . . . 4  |-  ( A. y A. z ( ( x A y  /\  x A z )  -> 
y  =  z )  <->  E. z A. y ( x A y  -> 
y  =  z ) )
76albii 1575 . . 3  |-  ( A. x A. y A. z
( ( x A y  /\  x A z )  ->  y  =  z )  <->  A. x E. z A. y ( x A y  -> 
y  =  z ) )
87anbi2i 676 . 2  |-  ( ( Rel  A  /\  A. x A. y A. z
( ( x A y  /\  x A z )  ->  y  =  z ) )  <-> 
( Rel  A  /\  A. x E. z A. y ( x A y  ->  y  =  z ) ) )
91, 8bitri 241 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E. z A. y ( x A y  ->  y  =  z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550   E*wmo 2281   class class class wbr 4204   Rel wrel 4875   Fun wfun 5440
This theorem is referenced by:  dffun5  5459  dffun6f  5460  dffv2  5788  sbcfun  27954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-cnv 4878  df-co 4879  df-fun 5448
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