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Theorem dfdm4 5066
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 2961 . . . . 5  |-  y  e. 
_V
2 vex 2961 . . . . 5  |-  x  e. 
_V
31, 2brcnv 5058 . . . 4  |-  ( y `' A x  <->  x A
y )
43exbii 1593 . . 3  |-  ( E. y  y `' A x 
<->  E. y  x A y )
54abbii 2550 . 2  |-  { x  |  E. y  y `' A x }  =  { x  |  E. y  x A y }
6 dfrn2 5062 . 2  |-  ran  `' A  =  { x  |  E. y  y `' A x }
7 df-dm 4891 . 2  |-  dom  A  =  { x  |  E. y  x A y }
85, 6, 73eqtr4ri 2469 1  |-  dom  A  =  ran  `' A
Colors of variables: wff set class
Syntax hints:   E.wex 1551    = wceq 1653   {cab 2424   class class class wbr 4215   `'ccnv 4880   dom cdm 4881   ran crn 4882
This theorem is referenced by:  dmcnvcnv  5095  rncnvcnv  5096  rncoeq  5142  cnvimass  5227  cnvimarndm  5228  dminxp  5314  cnvsn0  5341  rnsnopg  5352  dmmpt  5368  dmco  5381  cores2  5385  cnvssrndm  5394  unidmrn  5402  dfdm2  5404  cnvexg  5408  funimacnv  5528  foimacnv  5695  funcocnv2  5703  fimacnv  5865  f1opw2  6301  tz7.48-3  6704  fopwdom  7219  sbthlem4  7223  fodomr  7261  f1opwfi  7413  zorn2lem4  8384  unbenlem  13281  pjdm  16939  paste  17363  hmeores  17808  icchmeo  18971  gsumpropd2lem  24225  coinfliprv  24745  itg2addnclem2  26271  funsnfsup  26757  lnmlmic  27177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-cnv 4889  df-dm 4891  df-rn 4892
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