Theorem dffn3 6054

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 3624	. . 3 ⊢ ran 𝐹 ⊆ ran 𝐹
2	1	biantru 526	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹))
3		df-f 5892	. 2 ⊢ (𝐹:𝐴⟶ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ran 𝐹))
4	2, 3	bitr4i 267	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)

Syntax hints:   ↔ wb 196   ∧ wa 384   ⊆ wss 3574  ran crn 5115   Fn wfn 5883  ⟶wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-f 5892

This theorem is referenced by:  ffrn  6055  fsn2  6403  fo2ndf  7284  fndmfisuppfi  8287  fndmfifsupp  8288  fin23lem17  9160  fin23lem32  9166  fnct  9359  yoniso  16925  1stckgen  21357  ovolicc2  23290  itg1val2  23451  i1fadd  23462  i1fmul  23463  itg1addlem4  23466  i1fmulc  23470  clwlkclwwlklem2  26901  foresf1o  29343  fcoinver  29418  ofpreima2  29466  locfinreflem  29907  pl1cn  30001  poimirlem29  33438  poimirlem30  33439  itg2addnclem2  33462  mapdcl  36942  wessf1ornlem  39371  unirnmap  39400  fsneqrn  39403  icccncfext  40100  stoweidlem29  40246  stoweidlem31  40248  stoweidlem59  40276  subsaliuncllem  40575  meadjiunlem  40682