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Definition df-fun 4288
Description: Define predicate that determines if some class  A is a function. Definition 10.1 of [Quine] p. 65. For example, the expression  Fun  cos is true once we define cosine (df-cos 11555). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3650 with the maps-to notation (see df-mpt 3652 and df-mpt2 5375). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4289), a function with a given domain and codomain (df-f 4290), a one-to-one function (df-f1 4291), an onto function (df-fo 4292), or a one-to-one onto function (df-f1o 4293). For alternate definitions, see dffun2 4805, dffun3 4806, dffun4 4807, dffun5 4808, dffun6 4810, dffun7 4820, dffun8 4821, and dffun9 4822. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
df-fun  |-  ( Fun 
A  <->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  ) )

Detailed syntax breakdown of Definition df-fun
StepHypRef Expression
1 cA . . 3  class  A
21wfun 4272 . 2  wff  Fun  A
31wrel 4271 . . 3  wff  Rel  A
41ccnv 4265 . . . . 5  class  `' A
51, 4ccom 4270 . . . 4  class  ( A  o.  `' A )
6 cid 3865 . . . 4  class  _I
75, 6wss 2798 . . 3  wff  ( A  o.  `' A ) 
C_  _I
83, 7wa 356 . 2  wff  ( Rel 
A  /\  ( A  o.  `' A )  C_  _I  )
92, 8wb 174 1  wff  ( Fun 
A  <->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  ) )
Colors of variables: wff set class
This definition is referenced by:  dffun2  4805  funrel  4812  funss  4813  hbfun  4817  funi  4824  funcocnv2  5032  dffv2  5121
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