Theorem dffn2 5434

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 3217	. . 3 ⊢ ran 𝐹 ⊆ V
2	1	biantru 302	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
3		df-f 5281	. 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 2773   ⊆ wss 3168  ran crn 4681   Fn wfn 5272  ⟶wf 5273

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-v 2775  df-in 3174  df-ss 3181  df-f 5281

This theorem is referenced by:  f1cnvcnv  5501  fcoconst  5761  fnressn  5780  1stcof  6259  2ndcof  6260  fnmpo  6298  tposfn  6369  tfrlemibfn  6424  tfr1onlembfn  6440  mptelixpg  6831