HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version

Definition df-fun 4278
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 11535). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3640 with the maps-to notation (see df-mpt 3642 and df-mpt2 5365). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4279), a function with a given domain and codomain (df-f 4280), a one-to-one function (df-f1 4281), an onto function (df-fo 4282), or a one-to-one onto function (df-f1o 4283). For alternate definitions, see dffun2 4795, dffun3 4796, dffun4 4797, dffun5 4798, dffun6 4800, dffun7 4810, dffun8 4811, and dffun9 4812. (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 4262 . 2  wff  Fun  A
31wrel 4261 . . 3  wff  Rel  A
41ccnv 4255 . . . . 5  class  `' A
51, 4ccom 4260 . . . 4  class  ( A  o.  `' A )
6 cid 3855 . . . 4  class  _I
75, 6wss 2793 . . 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  4795  funrel  4802  funss  4803  hbfun  4807  funi  4814  funcocnv2  5022  dffv2  5111
Copyright terms: Public domain