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 𝐴 ↔ 𝐹:𝐴⟶V)

Proof of Theorem dffn2

Step	Hyp	Ref	Expression
1		ssv 3192	. . 3 ⊢ ran 𝐹 ⊆ V
2	1	biantru 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))
4	2, 3	bitr4i 187	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)

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