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Theorem dfdm4 4555
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 2605 . . . . 5  |-  y  e. 
_V
2 vex 2605 . . . . 5  |-  x  e. 
_V
31, 2brcnv 4546 . . . 4  |-  ( y `' A x  <->  x A
y )
43exbii 1537 . . 3  |-  ( E. y  y `' A x 
<->  E. y  x A y )
54abbii 2195 . 2  |-  { x  |  E. y  y `' A x }  =  { x  |  E. y  x A y }
6 dfrn2 4551 . 2  |-  ran  `' A  =  { x  |  E. y  y `' A x }
7 df-dm 4381 . 2  |-  dom  A  =  { x  |  E. y  x A y }
85, 6, 73eqtr4ri 2113 1  |-  dom  A  =  ran  `' A
Colors of variables: wff set class
Syntax hints:    = wceq 1285   E.wex 1422   {cab 2068   class class class wbr 3793   `'ccnv 4370   dom cdm 4371   ran crn 4372
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by:  dmcnvcnv  4586  rncnvcnv  4587  rncoeq  4633  cnvimass  4718  cnvimarndm  4719  dminxp  4795  cnvsn0  4819  rnsnopg  4829  dmmpt  4846  dmco  4859  cores2  4863  cnvssrndm  4872  unidmrn  4880  dfdm2  4882  cnvexg  4885  funimacnv  5006  foimacnv  5175  funcocnv2  5182  fimacnv  5328  f1opw2  5737  fopwdom  6380
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