Definition df-fun 5382

Description: Define predicate that determines if some class 𝐴 is a function. Definition 10.1 of [Quine] p. 65. For example, the expression Fun I is true (funi 5412). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4193 with the maps-to notation (see df-mpt 4195). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5383), a function with a given domain and codomain (df-f 5384), a one-to-one function (df-f1 5385), an onto function (df-fo 5386), or a one-to-one onto function (df-f1o 5387). For alternate definitions, see dffun2 5390, dffun4 5391, dffun6 5394, dffun7 5407, dffun8 5408, and dffun9 5409. (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 5374	. 2 wff Fun 𝐴
3	1	wrel 4781	. . 3 wff Rel 𝐴
4	1	ccnv 4775	. . . . 5 class ◡𝐴
5	1, 4	ccom 4780	. . . 4 class (𝐴 ∘ ◡𝐴)
6		cid 4435	. . . 4 class I
7	5, 6	wss 3220	. . 3 wff (𝐴 ∘ ◡𝐴) ⊆ I
8	3, 7	wa 104	. 2 wff (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )
9	2, 8	wb 105	1 wff (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))

This definition is used by:  dffun2  5390  funrel  5397  funss  5399  nffun  5403  funi  5412  funcocnv2  5667