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Theorem dffun8 5239
Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5238. (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 5238 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
2 df-mo 2030 . . . . 5  |-  ( E* y  x A y  <-> 
( E. y  x A y  ->  E! y  x A y ) )
3 vex 2740 . . . . . . 7  |-  x  e. 
_V
43eldm 4819 . . . . . 6  |-  ( x  e.  dom  A  <->  E. y  x A y )
5 pm5.5 242 . . . . . 6  |-  ( E. y  x A y  ->  ( ( E. y  x A y  ->  E! y  x A y )  <->  E! y  x A y ) )
64, 5sylbi 121 . . . . 5  |-  ( x  e.  dom  A  -> 
( ( E. y  x A y  ->  E! y  x A y )  <-> 
E! y  x A y ) )
72, 6bitrid 192 . . . 4  |-  ( x  e.  dom  A  -> 
( E* y  x A y  <->  E! y  x A y ) )
87ralbiia 2491 . . 3  |-  ( A. x  e.  dom  A E* y  x A y  <->  A. x  e.  dom  A E! y  x A y )
98anbi2i 457 . 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 184 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    /\ wa 104    <-> wb 105   E.wex 1492   E!weu 2026   E*wmo 2027    e. wcel 2148   A.wral 2455   class class class wbr 4000   dom cdm 4622   Rel wrel 4627   Fun wfun 5205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-id 4289  df-cnv 4630  df-co 4631  df-dm 4632  df-fun 5213
This theorem is referenced by:  funco  5251  funimaexglem  5294  funfveu  5523
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