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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 ). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt with the maps-to notation (see df-mpt and df-mpo ). Contrast this predicate with the predicates to determine if some class is a function with a given domain ( df-fn ), a function with a given domain and codomain ( df-f ), a one-to-one function ( df-f1 ), an onto function ( df-fo ), or a one-to-one onto function ( df-f1o ). For alternate definitions, see dffun2 , dffun3 , dffun4 , dffun5 , dffun6 , dffun7 , dffun8 , and dffun9 . (Contributed by NM, 1-Aug-1994)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-fun | |- ( Fun A <-> ( Rel A /\ ( A o. `' A ) C_ _I ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | cA | |- A |
|
| 1 | 0 | wfun | |- Fun A |
| 2 | 0 | wrel | |- Rel A |
| 3 | 0 | ccnv | |- `' A |
| 4 | 0 3 | ccom | |- ( A o. `' A ) |
| 5 | cid | |- _I |
|
| 6 | 4 5 | wss | |- ( A o. `' A ) C_ _I |
| 7 | 2 6 | wa | |- ( Rel A /\ ( A o. `' A ) C_ _I ) |
| 8 | 1 7 | wb | |- ( Fun A <-> ( Rel A /\ ( A o. `' A ) C_ _I ) ) |