MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-fun Unicode version

Definition df-fun 5257
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 12352). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4077 with the maps-to notation (see df-mpt 4079 and df-mpt2 5863). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5258), a function with a given domain and codomain (df-f 5259), a one-to-one function (df-f1 5260), an onto function (df-fo 5261), or a one-to-one onto function (df-f1o 5262). For alternate definitions, see dffun2 5265, dffun3 5266, dffun4 5267, dffun5 5268, dffun6 5270, dffun7 5280, dffun8 5281, and dffun9 5282. (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 5249 . 2  wff  Fun  A
31wrel 4694 . . 3  wff  Rel  A
41ccnv 4688 . . . . 5  class  `' A
51, 4ccom 4693 . . . 4  class  ( A  o.  `' A )
6 cid 4304 . . . 4  class  _I
75, 6wss 3152 . . 3  wff  ( A  o.  `' A ) 
C_  _I
83, 7wa 358 . 2  wff  ( Rel 
A  /\  ( A  o.  `' A )  C_  _I  )
92, 8wb 176 1  wff  ( Fun 
A  <->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  ) )
Colors of variables: wff set class
This definition is referenced by:  dffun2  5265  funrel  5272  funss  5273  nffun  5277  funi  5284  funcocnv2  5498  dffv2  5592
  Copyright terms: Public domain W3C validator