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Theorem dffun7 5296
Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one." However, dffun8 5297 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)
Assertion
Ref Expression
dffun7  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
Distinct variable group:    x, y, A

Proof of Theorem dffun7
StepHypRef Expression
1 dffun6 5286 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
2 moabs 2200 . . . . . 6  |-  ( E* y  x A y  <-> 
( E. y  x A y  ->  E* y  x A y ) )
3 vex 2804 . . . . . . . 8  |-  x  e. 
_V
43eldm 4892 . . . . . . 7  |-  ( x  e.  dom  A  <->  E. y  x A y )
54imbi1i 315 . . . . . 6  |-  ( ( x  e.  dom  A  ->  E* y  x A y )  <->  ( E. y  x A y  ->  E* y  x A
y ) )
62, 5bitr4i 243 . . . . 5  |-  ( E* y  x A y  <-> 
( x  e.  dom  A  ->  E* y  x A y ) )
76albii 1556 . . . 4  |-  ( A. x E* y  x A y  <->  A. x ( x  e.  dom  A  ->  E* y  x A
y ) )
8 df-ral 2561 . . . 4  |-  ( A. x  e.  dom  A E* y  x A y  <->  A. x
( x  e.  dom  A  ->  E* y  x A y ) )
97, 8bitr4i 243 . . 3  |-  ( A. x E* y  x A y  <->  A. x  e.  dom  A E* y  x A y )
109anbi2i 675 . 2  |-  ( ( Rel  A  /\  A. x E* y  x A y )  <->  ( Rel  A  /\  A. x  e. 
dom  A E* y  x A y ) )
111, 10bitri 240 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    e. wcel 1696   E*wmo 2157   A.wral 2556   class class class wbr 4039   dom cdm 4705   Rel wrel 4710   Fun wfun 5265
This theorem is referenced by:  dffun8  5297  dffun9  5298  brdom5  8170  imasaddfnlem  13446  imasvscafn  13455  funressnfv  28096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273
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