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Theorem dffn2 5386
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 3192 . . 3 ran 𝐹 ⊆ V
21biantru 302 . 2 (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
3 df-f 5239 . 2 (𝐹:𝐴⟶V ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
42, 3bitr4i 187 1 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  Vcvv 2752  wss 3144  ran crn 4645   Fn wfn 5230  wf 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754  df-in 3150  df-ss 3157  df-f 5239
This theorem is referenced by:  f1cnvcnv  5451  fcoconst  5708  fnressn  5723  1stcof  6188  2ndcof  6189  fnmpo  6227  tposfn  6298  tfrlemibfn  6353  tfr1onlembfn  6369  mptelixpg  6760
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