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| Mirrors > Home > MPE Home > Th. List > df-fun | Structured version Visualization version GIF version | ||
| 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 16106). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 5225 with the maps-to notation (see df-mpt 5226 and df-mpo 7436). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 6564), a function with a given domain and codomain (df-f 6565), a one-to-one function (df-f1 6566), an onto function (df-fo 6567), or a one-to-one onto function (df-f1o 6568). For alternate definitions, see dffun2 6571, dffun3 6575, dffun4 6577, dffun5 6578, dffun6 6574, dffun7 6593, dffun8 6594, and dffun9 6595. (Contributed by NM, 1-Aug-1994.) |
| Ref | Expression |
|---|---|
| df-fun | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cA | . . 3 class 𝐴 | |
| 2 | 1 | wfun 6555 | . 2 wff Fun 𝐴 |
| 3 | 1 | wrel 5690 | . . 3 wff Rel 𝐴 |
| 4 | 1 | ccnv 5684 | . . . . 5 class ◡𝐴 |
| 5 | 1, 4 | ccom 5689 | . . . 4 class (𝐴 ∘ ◡𝐴) |
| 6 | cid 5577 | . . . 4 class I | |
| 7 | 5, 6 | wss 3951 | . . 3 wff (𝐴 ∘ ◡𝐴) ⊆ I |
| 8 | 3, 7 | wa 395 | . 2 wff (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ) |
| 9 | 2, 8 | wb 206 | 1 wff (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) |
| Colors of variables: wff setvar class |
| This definition is referenced by: dffun2 6571 dffun2OLD 6572 dffun2OLDOLD 6573 funrel 6583 funss 6585 nffun 6589 funi 6598 funcocnv2 6873 dffv2 7004 funALTVfun 38699 |
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