Definition df-fun 5190

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 5220). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4043 with the maps-to notation (see df-mpt 4045). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5191), a function with a given domain and codomain (df-f 5192), a one-to-one function (df-f1 5193), an onto function (df-fo 5194), or a one-to-one onto function (df-f1o 5195). For alternate definitions, see dffun2 5198, dffun4 5199, dffun6 5202, dffun7 5215, dffun8 5216, and dffun9 5217. (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 5182	. 2 wff Fun 𝐴
3	1	wrel 4609	. . 3 wff Rel 𝐴
4	1	ccnv 4603	. . . . 5 class ◡𝐴
5	1, 4	ccom 4608	. . . 4 class (𝐴 ∘ ◡𝐴)
6		cid 4266	. . . 4 class I
7	5, 6	wss 3116	. . 3 wff (𝐴 ∘ ◡𝐴) ⊆ I
8	3, 7	wa 103	. 2 wff (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )
9	2, 8	wb 104	1 wff (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))

This definition is referenced by:  dffun2  5198  funrel  5205  funss  5207  nffun  5211  funi  5220  funcocnv2  5457