Definition df-fun 6382

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 15632). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5135 with the maps-to notation (see df-mpt 5136 and df-mpo 7218). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6383), a function with a given domain and codomain (df-f 6384), a one-to-one function (df-f1 6385), an onto function (df-fo 6386), or a one-to-one onto function (df-f1o 6387). For alternate definitions, see dffun2 6390, dffun3 6391, dffun4 6392, dffun5 6393, dffun6 6395, dffun7 6407, dffun8 6408, and dffun9 6409. (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 6374	. 2 wff Fun 𝐴
3	1	wrel 5556	. . 3 wff Rel 𝐴
4	1	ccnv 5550	. . . . 5 class ◡𝐴
5	1, 4	ccom 5555	. . . 4 class (𝐴 ∘ ◡𝐴)
6		cid 5454	. . . 4 class I
7	5, 6	wss 3866	. . 3 wff (𝐴 ∘ ◡𝐴) ⊆ I
8	3, 7	wa 399	. 2 wff (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )
9	2, 8	wb 209	1 wff (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))

This definition is referenced by:  dffun2  6390  funrel  6397  funss  6399  nffun  6403  funi  6412  funcocnv2  6685  dffv2  6806  funALTVfun  36546