Theorem fnima 6010

Description: The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fnima	|- ( F Fn A -> ( F " A ) = ran F )

Proof of Theorem fnima

Step	Hyp	Ref	Expression
1		df-ima 5127	. 2 |- ( F " A ) = ran ( F |` A )
2		fnresdm 6000	. . 3 |- ( F Fn A -> ( F |` A ) = F )
3	2	rneqd 5353	. 2 |- ( F Fn A -> ran ( F |` A ) = ran F )
4	1, 3	syl5eq 2668	1 |- ( F Fn A -> ( F " A ) = ran F )

Syntax hints:   -> wi 4   = wceq 1483   ran crn 5115   |` cres 5116   "cima 5117   Fn wfn 5883

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891

This theorem is referenced by:  infdifsn  8554  carduniima  8919  cardinfima  8920  alephfp  8931  dprdf1o  18431  dprd2db  18442  lmhmrnlss  19050  mpfsubrg  19532  pf1subrg  19712  frlmlbs  20136  frlmup3  20139  ellspd  20141  tgrest  20963  uniiccdif  23346  uniioombllem3  23353  dvgt0lem2  23766  eulerpartlemn  30443  matunitlindflem2  33406  poimirlem15  33424  k0004lem1  38445