Theorem dffn2 6733

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 3937	. . 3 ⊢ ran 𝐹 ⊆ V
2	1	biantru 539	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
3		df-f 6562	. 2 ⊢ (𝐹:𝐴⟶V ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
4	2, 3	bitr4i 281	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  Vcvv 3454   ⊆ wss 3900  ran crn 5656   Fn wfn 6553  ⟶wf 6554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-f 6562

This theorem is used by:  f1cnvcnv  6811  fcoconst  7159  fnressn  7186  fndifnfp  7204  1stcof  8055  2ndcof  8056  fnmpo  8105  tposfn  8300  tz7.48lem  8478  tz7.48lemOLD  8479  seqomlem2  8489  mptelixpg  8991  r111  9809  smobeth  10698  inar1  10887  imasvscafn  17738  fucidcl  18172  fucsect  18179  dfinito3  18209  dftermo3  18210  curfcl  18435  curf2ndf  18450  dsmmbas2  22073  frlmsslsp  22132  frlmup1  22134  prdstopn  23977  prdstps  23978  ist0-4  24078  ptuncnv  24156  xpstopnlem2  24160  prdstgpd  24474  prdsxmslem2  24878  curry2ima  33373  mplvrpmrhm  34249  acwer1prc  35852  onvf1od  36035  fnchoice  46251  fsneqrn  46429  stoweidlem35  47250  ixpv  50208  basresposfo  50297  fucorid2  50690  fucoridid  50691  precofval2  50709