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Theorem dffn2 5269
 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 3114 . . 3 ran 𝐹 ⊆ V
21biantru 300 . 2 (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
3 df-f 5122 . 2 (𝐹:𝐴⟶V ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
42, 3bitr4i 186 1 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104  Vcvv 2681   ⊆ wss 3066  ran crn 4535   Fn wfn 5113  ⟶wf 5114 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683  df-in 3072  df-ss 3079  df-f 5122 This theorem is referenced by:  f1cnvcnv  5334  fcoconst  5584  fnressn  5599  1stcof  6054  2ndcof  6055  fnmpo  6093  tposfn  6163  tfrlemibfn  6218  tfr1onlembfn  6234  mptelixpg  6621
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