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 5670
Description: Define the range of a class. For example, 𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9} (ex-rn 30906). Contrast with domain (defined in df-dm 5669). For alternate definitions, see dfrn2 5876, dfrn3 5877, and dfrn4 6200. 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 6070. Not to be confused with "codomain" (see df-f 6541), which may be a superset/superclass of the range (see frn 6714). (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 5660 . 2 class ran 𝐴
31ccnv 5658 . . 3 class 𝐴
43cdm 5659 . 2 class dom 𝐴
52, 4wceq 1570 1 wff ran 𝐴 = dom 𝐴
Colors of variables:    wff setvar class
This definition is used by:  dfrn2  5876  dmcnvcnv  5921  rncnvcnv  5922  rneq  5924  rnss  5927  brelrng  5929  nfrn  5940  rncoss  5965  rncoeq  5969  cnvimarndm  6083  rnun  6140  rninOLD  6142  rnxp  6167  rnxpss  6169  imainrect  6178  rnsnopg  6221  cnvssrndm  6272  unidmrn  6281  dfdm2  6283  funcnvpr  6599  funcnvtp  6600  funcnvqp  6601  fncnv  6610  funcnvres  6615  funimacnv  6618  fimacnvdisj  6757  dff1o4  6830  foimacnv  6839  funcocnv2  6847  f1ompt  7107  nvof1o  7284  cnvexg  7924  tz7.48-3  8436  errn  8722  omxpenlem  9079  sbthlem5  9092  sbthlem8  9095  sbthlem9  9096  fodomr  9129  domss2  9137  fodomfir  9300  rnfi  9310  zorn2lem4  10504  rnct  10531  fpwwe2lem12  10654  trclublem  15070  relexpcnv  15110  relexpnnrn  15120  invf  17861  cicsym  17897  cnvtsr  18680  znleval  21768  ordtbas2  23417  ordtcnv  23427  ordtrest2  23430  cnconn  23648  tgqtop  23939  adj1o  32361  fcoinver  33064  fresf1o  33091  fcnvgreu  33132  dfcnv2  33135  preiman0  33169  gsumhashmul  33494  cycpmfvlem  33539  cycpmfv1  33540  cycpmfv2  33541  cycpmfv3  33542  cycpmrn  33570  cnvordtrestixx  34410  xrge0iifhmeo  34433  mbfmcst  34757  0rrv  34949  elrn3  36328  dfrn6  39043  cnvresrn  39083  dmxrn  39122  dmcoss2  39279  cnvrcl0  44452  conrel2d  44491  relexpaddss  44545  rntrclfvRP  44558  ntrneifv2  44907
  Copyright terms: Public domain W3C validator