Theorem dffn2 5533

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 3270	. . 3 ⊢ ran 𝐹 ⊆ V
2	1	biantru 302	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
3		df-f 5379	. 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 2821   ⊆ wss 3220  ran crn 4773   Fn wfn 5370  ⟶wf 5371

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823  df-in 3226  df-ss 3233  df-f 5379

This theorem is referenced by:  f1cnvcnv  5607  fcoconst  5873  fnressn  5895  1stcof  6391  2ndcof  6392  fnmpo  6432  tposfn  6538  tfrlemibfn  6593  tfr1onlembfn  6609  mptelixpg  7010