Theorem dffn2 5351

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 3170	. . 3 ⊢ ran 𝐹 ⊆ V
2	1	biantru 300	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
3		df-f 5204	. 2 ⊢ (𝐹:𝐴⟶V ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
4	2, 3	bitr4i 186	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)

Syntax hints:   ∧ wa 103   ↔ wb 104  Vcvv 2731   ⊆ wss 3122  ran crn 4613   Fn wfn 5195  ⟶wf 5196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-11 1500  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-v 2733  df-in 3128  df-ss 3135  df-f 5204

This theorem is referenced by:  f1cnvcnv  5416  fcoconst  5671  fnressn  5686  1stcof  6146  2ndcof  6147  fnmpo  6185  tposfn  6256  tfrlemibfn  6311  tfr1onlembfn  6327  mptelixpg  6716