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 𝐴 ↔ 𝐹:𝐴⟶V)

Proof of Theorem dffn2

Step	Hyp	Ref	Expression
1		ssv 3044	. . 3 ⊢ ran 𝐹 ⊆ V
2	1	biantru 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))
4	2, 3	bitr4i 185	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)

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