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Theorem dfdm4 4737
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 2692 . . . . 5 𝑦 ∈ V
2 vex 2692 . . . . 5 𝑥 ∈ V
31, 2brcnv 4728 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
43exbii 1585 . . 3 (∃𝑦 𝑦𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦)
54abbii 2256 . 2 {𝑥 ∣ ∃𝑦 𝑦𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
6 dfrn2 4733 . 2 ran 𝐴 = {𝑥 ∣ ∃𝑦 𝑦𝐴𝑥}
7 df-dm 4555 . 2 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
85, 6, 73eqtr4ri 2172 1 dom 𝐴 = ran 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1332  wex 1469  {cab 2126   class class class wbr 3935  ccnv 4544  dom cdm 4545  ran crn 4546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-br 3936  df-opab 3996  df-cnv 4553  df-dm 4555  df-rn 4556
This theorem is referenced by:  dmcnvcnv  4769  rncnvcnv  4770  rncoeq  4818  cnvimass  4908  cnvimarndm  4909  dminxp  4989  cnvsn0  5013  rnsnopg  5023  dmmpt  5040  dmco  5053  cores2  5057  cnvssrndm  5066  cocnvres  5069  unidmrn  5077  dfdm2  5079  cnvexg  5082  funimacnv  5205  foimacnv  5391  funcocnv2  5398  fimacnv  5555  f1opw2  5982  fopwdom  6736  sbthlemi4  6854  exmidfodomrlemim  7072  hmeores  12516
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