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Theorem dfdm4 5852
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 3448 . . . . 5 𝑦 ∈ V
2 vex 3448 . . . . 5 𝑥 ∈ V
31, 2brcnv 5839 . . . 4 (𝑦𝐴𝑥𝑥𝐴𝑦)
43exbii 1851 . . 3 (∃𝑦 𝑦𝐴𝑥 ↔ ∃𝑦 𝑥𝐴𝑦)
54abbii 2803 . 2 {𝑥 ∣ ∃𝑦 𝑦𝐴𝑥} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
6 dfrn2 5845 . 2 ran 𝐴 = {𝑥 ∣ ∃𝑦 𝑦𝐴𝑥}
7 df-dm 5644 . 2 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
85, 6, 73eqtr4ri 2772 1 dom 𝐴 = ran 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wex 1782  {cab 2710   class class class wbr 5106  ccnv 5633  dom cdm 5634  ran crn 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  dmcnvcnv  5889  rncnvcnv  5890  rncoeq  5931  cnvimass  6034  cnvimarndm  6035  dminxp  6133  cnvsn0  6163  rnsnopg  6174  dmmpt  6193  dmco  6207  cores2  6212  cnvssrndm  6224  unidmrn  6232  dfdm2  6234  funimacnv  6583  foimacnv  6802  funcocnv2  6810  fimacnvOLD  7022  f1opw2  7609  cnvexg  7862  tz7.48-3  8391  fopwdom  9027  sbthlem4  9033  fodomr  9075  cnvfi  9127  f1opwfi  9303  zorn2lem4  10440  trclublem  14886  relexpcnv  14926  unbenlem  16785  gsumpropd2lem  18539  pjdm  21129  paste  22661  hmeores  23138  icchmeo  24320  fcnvgreu  31635  ffsrn  31693  gsummpt2co  31939  tocycfvres1  32008  tocycfvres2  32009  cycpmfvlem  32010  cycpmfv3  32013  coinfliprv  33139  itg2addnclem2  36176  rncnv  36807  lnmlmic  41458  dmnonrel  41950  cnvrcl0  41985  conrel1d  42023
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