Definition df-fun 6529

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 16203). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5185 with the maps-to notation (see df-mpt 5186 and df-mpo 7413). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6530), a function with a given domain and codomain (df-f 6531), a one-to-one function (df-f1 6532), an onto function (df-fo 6533), or a one-to-one onto function (df-f1o 6534). For alternate definitions, see dffun2 6537, dffun3 6539, dffun4 6540, dffun5 6541, dffun6 6538, dffun7 6555, dffun8 6556, and dffun9 6557. (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 6521	. 2 wff Fun 𝐴
3	1	wrel 5652	. . 3 wff Rel 𝐴
4	1	ccnv 5646	. . . . 5 class ◡𝐴
5	1, 4	ccom 5651	. . . 4 class (𝐴 ∘ ◡𝐴)
6		cid 5541	. . . 4 class I
7	5, 6	wss 3898	. . 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  6537  funrel  6544  funss  6546  nffun  6550  sbcfung  6551  funi  6560  funcocnv2  6838  dffv2  6968  nfchnd  18746  funALTVfun  39635