MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-rn Structured version   Visualization version   GIF version

Definition df-rn 5661
Description: Define the range of a class. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9} (ex-rn 31041). Contrast with domain (defined in df-dm 5660). For alternate definitions, see dfrn2 5869, dfrn3 5870, and dfrn4 6196. The notation "ran " is used by Enderton. The range of a function is often also called "the image of the function" (see definition in [Lang] p. ix), which can be justified by imadmrn 6066. Not to be confused with "codomain" (see df-f 6542), which may be a superset/superclass of the range (see frn 6717). (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
df-rn ran 𝐴 = dom ◡𝐴

Detailed syntax breakdown of Definition df-rn
StepHypRef Expression
1 cA . . 3 class 𝐴
21crn 5651 . 2 class ran 𝐴
31ccnv 5649 . . 3 class ◡𝐴
43cdm 5650 . 2 class dom ◡𝐴
52, 4wceq 1569 1 wff ran 𝐴 = dom ◡𝐴
Colors of variables:    wff setvar class
This definition is used by:  dfrn2  5869  dmcnvcnv  5914  rncnvcnv  5915  rneq  5917  rnss  5920  brelrng  5922  nfrn  5933  rncoss  5958  rncoeq  5962  cnvimassrndm  6078  cnvimarndmOLD  6080  rnun  6135  rninOLD  6137  rnxp  6161  rnxpss  6163  imainrect  6172  rnsnopg  6222  cnvssrndm  6273  unidmrn  6282  dfdm2  6284  funcnvpr  6602  funcnvtp  6603  funcnvqp  6604  fncnv  6613  funcnvres  6618  funimacnv  6621  fimacnvdisj  6760  dff1o4  6833  foimacnv  6842  funcocnv2  6850  f1ompt  7111  nvof1o  7288  cnvexg  7936  tz7.48-3  8454  errn  8740  omxpenlem  9097  sbthlem5  9110  sbthlem8  9113  sbthlem9  9114  fodomr  9147  domss2  9155  fodomfir  9319  rnfi  9329  zorn2lem4  10577  rnct  10604  fpwwe2lem12  10727  trclublem  15148  relexpcnv  15188  relexpnnrn  15198  invf  17943  cicsym  17979  cnvtsr  18762  znleval  21860  ordtbas2  23509  ordtcnv  23519  ordtrest2  23522  cnconn  23740  tgqtop  24031  adj1o  32496  fcoinver  33240  fresf1o  33267  fcnvgreu  33308  dfcnv2  33311  preiman0  33345  gsumhashmul  33670  cycpmfvlem  33715  cycpmfv1  33716  cycpmfv2  33717  cycpmfv3  33718  cycpmrn  33746  cnvordtrestixx  34587  xrge0iifhmeo  34610  mbfmcst  34933  0rrv  35125  elrn3  36569  dfrn6  39283  cnvresrn  39323  dmxrn  39362  dmcoss2  39519  cnvrcl0  44667  conrel2d  44706  relexpaddss  44760  rntrclfvRP  44773  ntrneifv2  45122  cocanss1  45966
  Copyright terms: Public domain W3C validator