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Theorem dfdm4 4616
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 2622 . . . . 5  |-  y  e. 
_V
2 vex 2622 . . . . 5  |-  x  e. 
_V
31, 2brcnv 4607 . . . 4  |-  ( y `' A x  <->  x A
y )
43exbii 1541 . . 3  |-  ( E. y  y `' A x 
<->  E. y  x A y )
54abbii 2203 . 2  |-  { x  |  E. y  y `' A x }  =  { x  |  E. y  x A y }
6 dfrn2 4612 . 2  |-  ran  `' A  =  { x  |  E. y  y `' A x }
7 df-dm 4438 . 2  |-  dom  A  =  { x  |  E. y  x A y }
85, 6, 73eqtr4ri 2119 1  |-  dom  A  =  ran  `' A
Colors of variables: wff set class
Syntax hints:    = wceq 1289   E.wex 1426   {cab 2074   class class class wbr 3837   `'ccnv 4427   dom cdm 4428   ran crn 4429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by:  dmcnvcnv  4647  rncnvcnv  4648  rncoeq  4694  cnvimass  4782  cnvimarndm  4783  dminxp  4862  cnvsn0  4886  rnsnopg  4896  dmmpt  4913  dmco  4926  cores2  4930  cnvssrndm  4939  cocnvres  4942  unidmrn  4950  dfdm2  4952  cnvexg  4955  funimacnv  5076  foimacnv  5255  funcocnv2  5262  fimacnv  5412  f1opw2  5832  fopwdom  6532  sbthlemi4  6648  exmidfodomrlemim  6806
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