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Theorem dfdm4 4872
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 2791 . . . . 5  |-  y  e. 
_V
2 vex 2791 . . . . 5  |-  x  e. 
_V
31, 2brcnv 4864 . . . 4  |-  ( y `' A x  <->  x A
y )
43exbii 1569 . . 3  |-  ( E. y  y `' A x 
<->  E. y  x A y )
54abbii 2395 . 2  |-  { x  |  E. y  y `' A x }  =  { x  |  E. y  x A y }
6 dfrn2 4868 . 2  |-  ran  `' A  =  { x  |  E. y  y `' A x }
7 df-dm 4699 . 2  |-  dom  A  =  { x  |  E. y  x A y }
85, 6, 73eqtr4ri 2314 1  |-  dom  A  =  ran  `' A
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623   {cab 2269   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ran crn 4690
This theorem is referenced by:  dmcnvcnv  4901  rncnvcnv  4902  rncoeq  4948  cnvimass  5033  cnvimarndm  5034  dminxp  5118  cnvsn0  5141  rnsnopg  5152  dmmpt  5168  dmco  5181  cores2  5185  cnvssrndm  5194  unidmrn  5202  dfdm2  5204  cnvexg  5208  funimacnv  5324  foimacnv  5490  funcocnv2  5498  fimacnv  5657  f1opw2  6071  tz7.48-3  6456  fopwdom  6970  sbthlem4  6974  fodomr  7012  f1opwfi  7159  zorn2lem4  8126  unbenlem  12955  pjdm  16607  paste  17022  hmeores  17462  icchmeo  18439  gsumpropd2lem  23379  coinfliprv  23683  fldcnv  25056  funsnfsup  26762  lnmlmic  27186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700
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