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Theorem dfdm4 4831
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 2752 . . . . 5  |-  y  e. 
_V
2 vex 2752 . . . . 5  |-  x  e. 
_V
31, 2brcnv 4822 . . . 4  |-  ( y `' A x  <->  x A
y )
43exbii 1615 . . 3  |-  ( E. y  y `' A x 
<->  E. y  x A y )
54abbii 2303 . 2  |-  { x  |  E. y  y `' A x }  =  { x  |  E. y  x A y }
6 dfrn2 4827 . 2  |-  ran  `' A  =  { x  |  E. y  y `' A x }
7 df-dm 4648 . 2  |-  dom  A  =  { x  |  E. y  x A y }
85, 6, 73eqtr4ri 2219 1  |-  dom  A  =  ran  `' A
Colors of variables: wff set class
Syntax hints:    = wceq 1363   E.wex 1502   {cab 2173   class class class wbr 4015   `'ccnv 4637   dom cdm 4638   ran crn 4639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-cnv 4646  df-dm 4648  df-rn 4649
This theorem is referenced by:  dmcnvcnv  4863  rncnvcnv  4864  rncoeq  4912  cnvimass  5003  cnvimarndm  5004  dminxp  5085  cnvsn0  5109  rnsnopg  5119  dmmpt  5136  dmco  5149  cores2  5153  cnvssrndm  5162  cocnvres  5165  unidmrn  5173  dfdm2  5175  cnvexg  5178  funimacnv  5304  foimacnv  5491  funcocnv2  5498  fimacnv  5658  f1opw2  6090  fopwdom  6849  sbthlemi4  6972  exmidfodomrlemim  7213  hmeores  14086
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