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Theorem dfdm4 5133
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 3074 . . . . 5  |-  y  e. 
_V
2 vex 3074 . . . . 5  |-  x  e. 
_V
31, 2brcnv 5123 . . . 4  |-  ( y `' A x  <->  x A
y )
43exbii 1635 . . 3  |-  ( E. y  y `' A x 
<->  E. y  x A y )
54abbii 2585 . 2  |-  { x  |  E. y  y `' A x }  =  { x  |  E. y  x A y }
6 dfrn2 5129 . 2  |-  ran  `' A  =  { x  |  E. y  y `' A x }
7 df-dm 4951 . 2  |-  dom  A  =  { x  |  E. y  x A y }
85, 6, 73eqtr4ri 2491 1  |-  dom  A  =  ran  `' A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   E.wex 1587   {cab 2436   class class class wbr 4393   `'ccnv 4940   dom cdm 4941   ran crn 4942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-cnv 4949  df-dm 4951  df-rn 4952
This theorem is referenced by:  dmcnvcnv  5163  rncnvcnv  5164  rncoeq  5204  cnvimass  5290  cnvimarndm  5291  dminxp  5379  cnvsn0  5408  rnsnopg  5419  dmmpt  5434  dmco  5447  cores2  5451  cnvssrndm  5460  unidmrn  5468  dfdm2  5470  funimacnv  5591  foimacnv  5759  funcocnv2  5766  fimacnv  5937  f1opw2  6416  cnvexg  6627  tz7.48-3  7002  fopwdom  7522  sbthlem4  7527  fodomr  7565  f1opwfi  7719  zorn2lem4  8772  unbenlem  14080  gsumpropd2lem  15616  funsnfsupOLD  17786  pjdm  18250  paste  19023  hmeores  19469  icchmeo  20638  fcnvgreu  26135  ffsrn  26173  gsummpt2co  26387  coinfliprv  27002  itg2addnclem2  28585  lnmlmic  29582
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