Definition df-fun 4827

Description: Define predicate that determines if some class A is a function. Definition 10.1 of [Quine] p. 65. For example, the expression Fun I is true (funi 4854). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3788 with the maps-to notation (see df-mpt 3790). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4828), a function with a given domain and codomain (df-f 4829), a one-to-one function (df-f1 4830), an onto function (df-fo 4831), or a one-to-one onto function (df-f1o 4832). For alternate definitions, see dffun2 4835, dffun4 4836, dffun6 4838, dffun7 4850, dffun8 4851, and dffun9 4852. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
df-fun	⊢ (Fun A ↔ (Rel A ∧ (A ∘ ◡A) ⊆ I ))

Detailed syntax breakdown of Definition df-fun

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2	1	wfun 4819	. 2 wff Fun A
3	1	wrel 4273	. . 3 wff Rel A
4	1	ccnv 4267	. . . . 5 class ◡A
5	1, 4	ccom 4272	. . . 4 class (A ∘ ◡A)
6		cid 3995	. . . 4 class I
7	5, 6	wss 2890	. . 3 wff (A ∘ ◡A) ⊆ I
8	3, 7	wa 97	. 2 wff (Rel A ∧ (A ∘ ◡A) ⊆ I )
9	2, 8	wb 98	1 wff (Fun A ↔ (Rel A ∧ (A ∘ ◡A) ⊆ I ))

This definition is referenced by:  dffun2  4835  funrel  4841  funss  4842  nffun  4846  funi  4854  funcocnv2  5072