Definition df-fun 6557

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 16262). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5189 with the maps-to notation (see df-mpt 5190 and df-mpo 7445). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6558), a function with a given domain and codomain (df-f 6559), a one-to-one function (df-f1 6560), an onto function (df-fo 6561), or a one-to-one onto function (df-f1o 6562). For alternate definitions, see dffun2 6565, dffun3 6567, dffun4 6568, dffun5 6569, dffun6 6566, dffun7 6584, dffun8 6585, and dffun9 6586. (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 6549	. 2 wff Fun 𝐴
3	1	wrel 5660	. . 3 wff Rel 𝐴
4	1	ccnv 5654	. . . . 5 class ◡𝐴
5	1, 4	ccom 5659	. . . 4 class (𝐴 ∘ ◡𝐴)
6		cid 5549	. . . 4 class I
7	5, 6	wss 3900	. . 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  6565  funrel  6573  funss  6575  nffun  6579  sbcfung  6580  funi  6589  funcocnv2  6869  funcocnv2OLD  6870  dffv2  7000  nfchnd  18811  funALTVfun  39906