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Theorem dffn2 5244
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 3089 . . 3 ran 𝐹 ⊆ V
21biantru 300 . 2 (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
3 df-f 5097 . 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 2660  wss 3041  ran crn 4510   Fn wfn 5088  wf 5089
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662  df-in 3047  df-ss 3054  df-f 5097
This theorem is referenced by:  f1cnvcnv  5309  fcoconst  5559  fnressn  5574  1stcof  6029  2ndcof  6030  fnmpo  6068  tposfn  6138  tfrlemibfn  6193  tfr1onlembfn  6209  mptelixpg  6596
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