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Theorem dfdm4 4701
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 2663 . . . . 5  |-  y  e. 
_V
2 vex 2663 . . . . 5  |-  x  e. 
_V
31, 2brcnv 4692 . . . 4  |-  ( y `' A x  <->  x A
y )
43exbii 1569 . . 3  |-  ( E. y  y `' A x 
<->  E. y  x A y )
54abbii 2233 . 2  |-  { x  |  E. y  y `' A x }  =  { x  |  E. y  x A y }
6 dfrn2 4697 . 2  |-  ran  `' A  =  { x  |  E. y  y `' A x }
7 df-dm 4519 . 2  |-  dom  A  =  { x  |  E. y  x A y }
85, 6, 73eqtr4ri 2149 1  |-  dom  A  =  ran  `' A
Colors of variables: wff set class
Syntax hints:    = wceq 1316   E.wex 1453   {cab 2103   class class class wbr 3899   `'ccnv 4508   dom cdm 4509   ran crn 4510
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by:  dmcnvcnv  4733  rncnvcnv  4734  rncoeq  4782  cnvimass  4872  cnvimarndm  4873  dminxp  4953  cnvsn0  4977  rnsnopg  4987  dmmpt  5004  dmco  5017  cores2  5021  cnvssrndm  5030  cocnvres  5033  unidmrn  5041  dfdm2  5043  cnvexg  5046  funimacnv  5169  foimacnv  5353  funcocnv2  5360  fimacnv  5517  f1opw2  5944  fopwdom  6698  sbthlemi4  6816  exmidfodomrlemim  7025  hmeores  12411
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