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Theorem dfdm4 5519
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 3388 . . . . 5 𝑦 ∈ V
2 vex 3388 . . . . 5 𝑥 ∈ V
31, 2brcnv 5508 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
43exbii 1944 . . 3 (∃𝑦 𝑦𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦)
54abbii 2916 . 2 {𝑥 ∣ ∃𝑦 𝑦𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
6 dfrn2 5514 . 2 ran 𝐴 = {𝑥 ∣ ∃𝑦 𝑦𝐴𝑥}
7 df-dm 5322 . 2 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
85, 6, 73eqtr4ri 2832 1 dom 𝐴 = ran 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wex 1875  {cab 2785   class class class wbr 4843  ccnv 5311  dom cdm 5312  ran crn 5313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-cnv 5320  df-dm 5322  df-rn 5323
This theorem is referenced by:  dmcnvcnv  5551  rncnvcnv  5552  rncoeq  5593  cnvimass  5702  cnvimarndm  5703  dminxp  5791  cnvsn0  5819  rnsnopg  5831  dmmpt  5849  dmco  5862  cores2  5867  cnvssrndm  5876  unidmrn  5884  dfdm2  5886  funimacnv  6181  foimacnv  6373  funcocnv2  6380  fimacnv  6573  f1opw2  7122  cnvexg  7347  tz7.48-3  7778  fopwdom  8310  sbthlem4  8315  fodomr  8353  f1opwfi  8512  zorn2lem4  9609  trclublem  14077  relexpcnv  14116  unbenlem  15945  gsumpropd2lem  17588  pjdm  20376  paste  21427  hmeores  21903  icchmeo  23068  fcnvgreu  29990  ffsrn  30022  gsummpt2co  30296  coinfliprv  31061  itg2addnclem2  33950  rncnv  34566  lnmlmic  38443  dmnonrel  38679  cnvrcl0  38715  conrel1d  38738
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