Theorem dffn3 6658

Description: A function maps to its range. (Contributed by NM, 1-Sep-1999.)

Assertion

Ref	Expression
dffn3	⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)

Proof of Theorem dffn3

Step	Hyp	Ref	Expression
1		ssid 3953	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2	1	biantru 530	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹))
3		df-f 6477	. 2 ⊢ (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹))
4	2, 3	bitr4i 277	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)

Syntax hints:   ↔ wb 205   ∧ wa 396   ⊆ wss 3897  ran crn 5615   Fn wfn 6468  ⟶wf 6469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-in 3904  df-ss 3914  df-f 6477

This theorem is referenced by:  ffrn  6659  ffrnb  6660  fsn2  7058  offsplitfpar  8019  fo2ndf  8021  suppcoss  8085  fndmfisuppfi  9230  fndmfifsupp  9231  fin23lem17  10187  fin23lem32  10193  fnct  10386  yoniso  18092  1stckgen  22803  ovolicc2  24784  i1fadd  24957  i1fmul  24958  itg1addlem4  24961  itg1addlem4OLD  24962  i1fmulc  24966  clwlkclwwlklem2  28593  foresf1o  31079  fcoinver  31174  ofpreima2  31231  locfinreflem  32029  pl1cn  32144  fvineqsneu  35680  poimirlem29  35904  poimirlem30  35905  itg2addnclem2  35927  mapdcl  39914  wessf1ornlem  43038  unirnmap  43064  fsneqrn  43067  icccncfext  43753  stoweidlem29  43895  stoweidlem31  43897  stoweidlem59  43925  subsaliuncllem  44221  meadjiunlem  44329  uniimaprimaeqfv  45174  uniimaelsetpreimafv  45188