Definition df-fun 4904

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 4932). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3818 with the maps-to notation (see df-mpt 3820). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 4905), a function with a given domain and codomain (df-f 4906), a one-to-one function (df-f1 4907), an onto function (df-fo 4908), or a one-to-one onto function (df-f1o 4909). For alternate definitions, see dffun2 4912, dffun4 4913, dffun6 4916, dffun7 4928, dffun8 4929, and dffun9 4930. (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 4896	. 2 wff Fun 𝐴
3	1	wrel 4350	. . 3 wff Rel 𝐴
4	1	ccnv 4344	. . . . 5 class ◡𝐴
5	1, 4	ccom 4349	. . . 4 class (𝐴 ∘ ◡𝐴)
6		cid 4025	. . . 4 class I
7	5, 6	wss 2917	. . 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  4912  funrel  4919  funss  4920  nffun  4924  funi  4932  funcocnv2  5151