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Definition df-fun 5555
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 12793). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4351 with the maps-to notation (see df-mpt 4353 and df-mpt2 6186). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5556), a function with a given domain and codomain (df-f 5557), a one-to-one function (df-f1 5558), an onto function (df-fo 5559), or a one-to-one onto function (df-f1o 5560). For alternate definitions, see dffun2 5563, dffun3 5564, dffun4 5565, dffun5 5566, dffun6 5568, dffun7 5578, dffun8 5579, and dffun9 5580. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
df-fun (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))

Detailed syntax breakdown of Definition df-fun
StepHypRef Expression
1 cA . . 3 class 𝐴
21wfun 5547 . 2 wff Fun 𝐴
31wrel 4975 . . 3 wff Rel 𝐴
41ccnv 4969 . . . . 5 class 𝐴
51, 4ccom 4974 . . . 4 class (𝐴𝐴)
6 cid 4585 . . . 4 class I
75, 6wss 3350 . . 3 wff (𝐴𝐴) ⊆ I
83, 7wa 360 . 2 wff (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I )
92, 8wb 178 1 wff (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
Colors of variables: wff set class
This definition is referenced by:  dffun2  5563  funrel  5570  funss  5571  nffun  5575  funi  5582  funcocnv2  5799  dffv2  5896
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