Definition df-fun 6553

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 16052). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5233 with the maps-to notation (see df-mpt 5234 and df-mpo 7429). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6554), a function with a given domain and codomain (df-f 6555), a one-to-one function (df-f1 6556), an onto function (df-fo 6557), or a one-to-one onto function (df-f1o 6558). For alternate definitions, see dffun2 6561, dffun3 6565, dffun4 6567, dffun5 6568, dffun6 6564, dffun7 6583, dffun8 6584, and dffun9 6585. (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 6545	. 2 wff Fun 𝐴
3	1	wrel 5685	. . 3 wff Rel 𝐴
4	1	ccnv 5679	. . . . 5 class ◡𝐴
5	1, 4	ccom 5684	. . . 4 class (𝐴 ∘ ◡𝐴)
6		cid 5577	. . . 4 class I
7	5, 6	wss 3947	. . 3 wff (𝐴 ∘ ◡𝐴) ⊆ I
8	3, 7	wa 394	. 2 wff (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )
9	2, 8	wb 205	1 wff (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))

This definition is referenced by:  dffun2  6561  dffun2OLD  6562  dffun2OLDOLD  6563  funrel  6573  funss  6575  nffun  6579  funi  6588  funcocnv2  6867  dffv2  6996  funALTVfun  38174