Theorem dffn2 6047

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 3625	. . 3 ⊢ ran 𝐹 ⊆ V
2	1	biantru 526	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
3		df-f 5892	. 2 ⊢ (𝐹:𝐴⟶V ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
4	2, 3	bitr4i 267	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)

Syntax hints:   ↔ wb 196   ∧ wa 384  Vcvv 3200   ⊆ wss 3574  ran crn 5115   Fn wfn 5883  ⟶wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-in 3581  df-ss 3588  df-f 5892

This theorem is referenced by:  f1cnvcnv  6109  fcoconst  6401  fnressn  6425  fndifnfp  6442  1stcof  7196  2ndcof  7197  fnmpt2  7238  tposfn  7381  tz7.48lem  7536  seqomlem2  7546  mptelixpg  7945  r111  8638  smobeth  9408  inar1  9597  imasvscafn  16197  fucidcl  16625  fucsect  16632  curfcl  16872  curf2ndf  16887  dsmmbas2  20081  frlmsslsp  20135  frlmup1  20137  prdstopn  21431  prdstps  21432  ist0-4  21532  ptuncnv  21610  xpstopnlem2  21614  prdstgpd  21928  prdsxmslem2  22334  curry2ima  29486  fnchoice  39188  fsneqrn  39403  stoweidlem35  40252