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Theorem dffun8 5620
Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5619. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
dffun8  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E! y  x A y ) )
Distinct variable group:    x, y, A

Proof of Theorem dffun8
StepHypRef Expression
1 dffun7 5619 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
2 df-mo 2268 . . . . 5  |-  ( E* y  x A y  <-> 
( E. y  x A y  ->  E! y  x A y ) )
3 vex 3081 . . . . . . 7  |-  x  e. 
_V
43eldm 5044 . . . . . 6  |-  ( x  e.  dom  A  <->  E. y  x A y )
5 pm5.5 337 . . . . . 6  |-  ( E. y  x A y  ->  ( ( E. y  x A y  ->  E! y  x A y )  <->  E! y  x A y ) )
64, 5sylbi 198 . . . . 5  |-  ( x  e.  dom  A  -> 
( ( E. y  x A y  ->  E! y  x A y )  <-> 
E! y  x A y ) )
72, 6syl5bb 260 . . . 4  |-  ( x  e.  dom  A  -> 
( E* y  x A y  <->  E! y  x A y ) )
87ralbiia 2853 . . 3  |-  ( A. x  e.  dom  A E* y  x A y  <->  A. x  e.  dom  A E! y  x A y )
98anbi2i 698 . 2  |-  ( ( Rel  A  /\  A. x  e.  dom  A E* y  x A y )  <-> 
( Rel  A  /\  A. x  e.  dom  A E! y  x A
y ) )
101, 9bitri 252 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E! y  x A y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   E.wex 1659    e. wcel 1867   E!weu 2263   E*wmo 2264   A.wral 2773   class class class wbr 4417   dom cdm 4846   Rel wrel 4851   Fun wfun 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4477  df-id 4761  df-cnv 4854  df-co 4855  df-dm 4856  df-fun 5595
This theorem is referenced by:  dfdfat2  38253
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