Definition df-fun 6563

Description: Define predicate that determines if some class 𝐴 is a function. Definition 10.1 of [Quine] p. 65. For example, the expression Fun cos is true once we define cosine (df-cos 16106). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5225 with the maps-to notation (see df-mpt 5226 and df-mpo 7436). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6564), a function with a given domain and codomain (df-f 6565), a one-to-one function (df-f1 6566), an onto function (df-fo 6567), or a one-to-one onto function (df-f1o 6568). For alternate definitions, see dffun2 6571, dffun3 6575, dffun4 6577, dffun5 6578, dffun6 6574, dffun7 6593, dffun8 6594, and dffun9 6595. (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 6555	. 2 wff Fun 𝐴
3	1	wrel 5690	. . 3 wff Rel 𝐴
4	1	ccnv 5684	. . . . 5 class ◡𝐴
5	1, 4	ccom 5689	. . . 4 class (𝐴 ∘ ◡𝐴)
6		cid 5577	. . . 4 class I
7	5, 6	wss 3951	. . 3 wff (𝐴 ∘ ◡𝐴) ⊆ I
8	3, 7	wa 395	. 2 wff (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )
9	2, 8	wb 206	1 wff (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))

This definition is referenced by:  dffun2  6571  dffun2OLD  6572  dffun2OLDOLD  6573  funrel  6583  funss  6585  nffun  6589  funi  6598  funcocnv2  6873  dffv2  7004  funALTVfun  38699