Definition df-fun 6539

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 16160). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5190 with the maps-to notation (see df-mpt 5191 and df-mpo 7421). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6540), a function with a given domain and codomain (df-f 6541), a one-to-one function (df-f1 6542), an onto function (df-fo 6543), or a one-to-one onto function (df-f1o 6544). For alternate definitions, see dffun2 6547, dffun3 6549, dffun4 6550, dffun5 6551, dffun6 6548, dffun7 6564, dffun8 6565, and dffun9 6566. (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 6531	. 2 wff Fun 𝐴
3	1	wrel 5664	. . 3 wff Rel 𝐴
4	1	ccnv 5658	. . . . 5 class ◡𝐴
5	1, 4	ccom 5663	. . . 4 class (𝐴 ∘ ◡𝐴)
6		cid 5553	. . . 4 class I
7	5, 6	wss 3902	. . 3 wff (𝐴 ∘ ◡𝐴) ⊆ I
8	3, 7	wa 401	. 2 wff (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )
9	2, 8	wb 209	1 wff (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))

This definition is used by:  dffun2  6547  funrel  6554  funss  6556  nffun  6560  funi  6569  funcocnv2  6847  dffv2  6977  nfchnd  18703  funALTVfun  39518