Definition df-fun 6575

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 16118). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5249 with the maps-to notation (see df-mpt 5250 and df-mpo 7453). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6576), a function with a given domain and codomain (df-f 6577), a one-to-one function (df-f1 6578), an onto function (df-fo 6579), or a one-to-one onto function (df-f1o 6580). For alternate definitions, see dffun2 6583, dffun3 6587, dffun4 6589, dffun5 6590, dffun6 6586, dffun7 6605, dffun8 6606, and dffun9 6607. (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 6567	. 2 wff Fun 𝐴
3	1	wrel 5705	. . 3 wff Rel 𝐴
4	1	ccnv 5699	. . . . 5 class ◡𝐴
5	1, 4	ccom 5704	. . . 4 class (𝐴 ∘ ◡𝐴)
6		cid 5592	. . . 4 class I
7	5, 6	wss 3976	. . 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  6583  dffun2OLD  6584  dffun2OLDOLD  6585  funrel  6595  funss  6597  nffun  6601  funi  6610  funcocnv2  6887  dffv2  7017  funALTVfun  38654