Definition df-fun 6543

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 16011). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5231 with the maps-to notation (see df-mpt 5232 and df-mpo 7411). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6544), a function with a given domain and codomain (df-f 6545), a one-to-one function (df-f1 6546), an onto function (df-fo 6547), or a one-to-one onto function (df-f1o 6548). For alternate definitions, see dffun2 6551, dffun3 6555, dffun4 6557, dffun5 6558, dffun6 6554, dffun7 6573, dffun8 6574, and dffun9 6575. (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 6535	. 2 wff Fun 𝐴
3	1	wrel 5681	. . 3 wff Rel 𝐴
4	1	ccnv 5675	. . . . 5 class ◡𝐴
5	1, 4	ccom 5680	. . . 4 class (𝐴 ∘ ◡𝐴)
6		cid 5573	. . . 4 class I
7	5, 6	wss 3948	. . 3 wff (𝐴 ∘ ◡𝐴) ⊆ I
8	3, 7	wa 397	. 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  6551  dffun2OLD  6552  dffun2OLDOLD  6553  funrel  6563  funss  6565  nffun  6569  funi  6578  funcocnv2  6856  dffv2  6984  funALTVfun  37557