ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dffn2 GIF version

Theorem dffn2 5347
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 𝐴𝐹:𝐴⟶V)

Proof of Theorem dffn2
StepHypRef Expression
1 ssv 3169 . . 3 ran 𝐹 ⊆ V
21biantru 300 . 2 (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
3 df-f 5200 . 2 (𝐹:𝐴⟶V ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ V))
42, 3bitr4i 186 1 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  Vcvv 2730  wss 3121  ran crn 4610   Fn wfn 5191  wf 5192
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-in 3127  df-ss 3134  df-f 5200
This theorem is referenced by:  f1cnvcnv  5412  fcoconst  5665  fnressn  5680  1stcof  6140  2ndcof  6141  fnmpo  6179  tposfn  6250  tfrlemibfn  6305  tfr1onlembfn  6321  mptelixpg  6710
  Copyright terms: Public domain W3C validator