Definition df-fun 6513

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 16036). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5188 with the maps-to notation (see df-mpt 5189 and df-mpo 7392). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6514), a function with a given domain and codomain (df-f 6515), a one-to-one function (df-f1 6516), an onto function (df-fo 6517), or a one-to-one onto function (df-f1o 6518). For alternate definitions, see dffun2 6521, dffun3 6525, dffun4 6527, dffun5 6528, dffun6 6524, dffun7 6543, dffun8 6544, and dffun9 6545. (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 6505	. 2 wff Fun 𝐴
3	1	wrel 5643	. . 3 wff Rel 𝐴
4	1	ccnv 5637	. . . . 5 class ◡𝐴
5	1, 4	ccom 5642	. . . 4 class (𝐴 ∘ ◡𝐴)
6		cid 5532	. . . 4 class I
7	5, 6	wss 3914	. . 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  6521  dffun2OLD  6522  dffun2OLDOLD  6523  funrel  6533  funss  6535  nffun  6539  funi  6548  funcocnv2  6825  dffv2  6956  funALTVfun  38690