Definition df-fun 4847

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 4875). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3809 with the maps-to notation (see df-mpt 3811). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4848), a function with a given domain and codomain (df-f 4849), a one-to-one function (df-f1 4850), an onto function (df-fo 4851), or a one-to-one onto function (df-f1o 4852). For alternate definitions, see dffun2 4855, dffun4 4856, dffun6 4859, dffun7 4871, dffun8 4872, and dffun9 4873. (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 4839	. 2 wff Fun A
3	1	wrel 4293	. . 3 wff Rel A
4	1	ccnv 4287	. . . . 5 class ◡A
5	1, 4	ccom 4292	. . . 4 class (A ∘ ◡A)
6		cid 4016	. . . 4 class I
7	5, 6	wss 2911	. . 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  4855  funrel  4862  funss  4863  nffun  4867  funi  4875  funcocnv2  5094