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Definition df-fun 5485
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 12704). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4291 with the maps-to notation (see df-mpt 4293 and df-mpt2 6115). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5486), a function with a given domain and codomain (df-f 5487), a one-to-one function (df-f1 5488), an onto function (df-fo 5489), or a one-to-one onto function (df-f1o 5490). For alternate definitions, see dffun2 5493, dffun3 5494, dffun4 5495, dffun5 5496, dffun6 5498, dffun7 5508, dffun8 5509, and dffun9 5510. (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 5477 . 2  wff  Fun  A
31wrel 4912 . . 3  wff  Rel  A
41ccnv 4906 . . . . 5  class  `' A
51, 4ccom 4911 . . . 4  class  ( A  o.  `' A )
6 cid 4522 . . . 4  class  _I
75, 6wss 3306 . . 3  wff  ( A  o.  `' A ) 
C_  _I
83, 7wa 360 . 2  wff  ( Rel 
A  /\  ( A  o.  `' A )  C_  _I  )
92, 8wb 178 1  wff  ( Fun 
A  <->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  ) )
Colors of variables: wff set class
This definition is referenced by:  dffun2  5493  funrel  5500  funss  5501  nffun  5505  funi  5512  funcocnv2  5729  dffv2  5825
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