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Theorem dffun8 5247
Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5246. (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 5246 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
2 df-mo 2149 . . . . 5  |-  ( E* y  x A y  <-> 
( E. y  x A y  ->  E! y  x A y ) )
3 vex 2792 . . . . . . 7  |-  x  e. 
_V
43eldm 4875 . . . . . 6  |-  ( x  e.  dom  A  <->  E. y  x A y )
5 pm5.5 328 . . . . . 6  |-  ( E. y  x A y  ->  ( ( E. y  x A y  ->  E! y  x A y )  <->  E! y  x A y ) )
64, 5sylbi 189 . . . . 5  |-  ( x  e.  dom  A  -> 
( ( E. y  x A y  ->  E! y  x A y )  <-> 
E! y  x A y ) )
72, 6syl5bb 250 . . . 4  |-  ( x  e.  dom  A  -> 
( E* y  x A y  <->  E! y  x A y ) )
87ralbiia 2576 . . 3  |-  ( A. x  e.  dom  A E* y  x A y  <->  A. x  e.  dom  A E! y  x A y )
98anbi2i 678 . 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 242 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E! y  x A y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   E.wex 1533    e. wcel 1688   E!weu 2144   E*wmo 2145   A.wral 2544   class class class wbr 4024   dom cdm 4688   Rel wrel 4693   Fun wfun 5215
This theorem is referenced by:  dfdfat2  27369
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-id 4308  df-cnv 4696  df-co 4697  df-dm 4698  df-fun 5223
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