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Theorem dffn2 5149
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 3044 . . 3 ran 𝐹 ⊆ V
21biantru 296 . 2 (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
3 df-f 5006 . 2 (𝐹:𝐴⟶V ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
42, 3bitr4i 185 1 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  Vcvv 2619  wss 2997  ran crn 4429   Fn wfn 4997  wf 4998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-in 3003  df-ss 3010  df-f 5006
This theorem is referenced by:  f1cnvcnv  5211  fcoconst  5452  fnressn  5467  1stcof  5916  2ndcof  5917  fnmpt2  5954  tposfn  6020  tfrlemibfn  6075  tfr1onlembfn  6091
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