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Theorem dffn2 5047
 Description: Any function is a mapping into V. (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
dffn2 (𝐹 Fn 𝐴𝐹:𝐴⟶V)

Proof of Theorem dffn2
StepHypRef Expression
1 ssv 2965 . . 3 ran 𝐹 ⊆ V
21biantru 286 . 2 (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
3 df-f 4906 . 2 (𝐹:𝐴⟶V ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
42, 3bitr4i 176 1 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  Vcvv 2557   ⊆ wss 2917  ran crn 4346   Fn wfn 4897  ⟶wf 4898 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559  df-in 2924  df-ss 2931  df-f 4906 This theorem is referenced by:  f1cnvcnv  5100  fcoconst  5334  fnressn  5349  1stcof  5790  2ndcof  5791  fnmpt2  5828  tposfn  5888  tfrlemibfn  5942
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