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Theorem dfdm4 4833
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 2754 . . . . 5 𝑦 ∈ V
2 vex 2754 . . . . 5 𝑥 ∈ V
31, 2brcnv 4824 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
43exbii 1615 . . 3 (∃𝑦 𝑦𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦)
54abbii 2304 . 2 {𝑥 ∣ ∃𝑦 𝑦𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
6 dfrn2 4829 . 2 ran 𝐴 = {𝑥 ∣ ∃𝑦 𝑦𝐴𝑥}
7 df-dm 4650 . 2 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
85, 6, 73eqtr4ri 2220 1 dom 𝐴 = ran 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1363  wex 1502  {cab 2174   class class class wbr 4017  ccnv 4639  dom cdm 4640  ran crn 4641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-br 4018  df-opab 4079  df-cnv 4648  df-dm 4650  df-rn 4651
This theorem is referenced by:  dmcnvcnv  4865  rncnvcnv  4866  rncoeq  4914  cnvimass  5005  cnvimarndm  5006  dminxp  5087  cnvsn0  5111  rnsnopg  5121  dmmpt  5138  dmco  5151  cores2  5155  cnvssrndm  5164  cocnvres  5167  unidmrn  5175  dfdm2  5177  cnvexg  5180  funimacnv  5306  foimacnv  5493  funcocnv2  5500  fimacnv  5660  f1opw2  6094  fopwdom  6853  sbthlemi4  6976  exmidfodomrlemim  7217  hmeores  14198
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