Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfdm4 Structured version   Visualization version   GIF version

Theorem dfdm4 5238
 Description: Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
Assertion
Ref Expression
dfdm4 dom 𝐴 = ran 𝐴

Proof of Theorem dfdm4
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3176 . . . . 5 𝑦 ∈ V
2 vex 3176 . . . . 5 𝑥 ∈ V
31, 2brcnv 5227 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
43exbii 1764 . . 3 (∃𝑦 𝑦𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦)
54abbii 2726 . 2 {𝑥 ∣ ∃𝑦 𝑦𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
6 dfrn2 5233 . 2 ran 𝐴 = {𝑥 ∣ ∃𝑦 𝑦𝐴𝑥}
7 df-dm 5048 . 2 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
85, 6, 73eqtr4ri 2643 1 dom 𝐴 = ran 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∃wex 1695  {cab 2596   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  ran crn 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by:  dmcnvcnv  5269  rncnvcnv  5270  rncoeq  5310  cnvimass  5404  cnvimarndm  5405  dminxp  5493  cnvsn0  5521  rnsnopg  5532  dmmpt  5547  dmco  5560  cores2  5565  cnvssrndm  5574  unidmrn  5582  dfdm2  5584  funimacnv  5884  foimacnv  6067  funcocnv2  6074  fimacnv  6255  f1opw2  6786  cnvexg  7005  tz7.48-3  7426  fopwdom  7953  sbthlem4  7958  fodomr  7996  f1opwfi  8153  zorn2lem4  9204  trclublem  13582  relexpcnv  13623  unbenlem  15450  gsumpropd2lem  17096  pjdm  19870  paste  20908  hmeores  21384  icchmeo  22548  fcnvgreu  28855  ffsrn  28892  gsummpt2co  29111  coinfliprv  29871  itg2addnclem2  32632  lnmlmic  36676  dmnonrel  36915  cnvrcl0  36951  conrel1d  36974
 Copyright terms: Public domain W3C validator