Definition df-fun 4822

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 4849). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3784 with the maps-to notation (see df-mpt 3786). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4823), a function with a given domain and codomain (df-f 4824), a one-to-one function (df-f1 4825), an onto function (df-fo 4826), or a one-to-one onto function (df-f1o 4827). For alternate definitions, see dffun2 4830, dffun4 4831, dffun6 4833, dffun7 4845, dffun8 4846, and dffun9 4847. (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 4814	. 2 wff Fun A
3	1	wrel 4268	. . 3 wff Rel A
4	1	ccnv 4262	. . . . 5 class ◡A
5	1, 4	ccom 4267	. . . 4 class (A ∘ ◡A)
6		cid 3991	. . . 4 class I
7	5, 6	wss 2888	. . 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  4830  funrel  4836  funss  4837  nffun  4841  funi  4849  funcocnv2  5067