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Theorem dfdm4 4470
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 2554 . . . . 5 y V
2 vex 2554 . . . . 5 x V
31, 2brcnv 4461 . . . 4 (yAxxAy)
43exbii 1493 . . 3 (y yAxy xAy)
54abbii 2150 . 2 {xy yAx} = {xy xAy}
6 dfrn2 4466 . 2 ran A = {xy yAx}
7 df-dm 4298 . 2 dom A = {xy xAy}
85, 6, 73eqtr4ri 2068 1 dom A = ran A
Colors of variables: wff set class
Syntax hints:   = wceq 1242  wex 1378  {cab 2023   class class class wbr 3755  ccnv 4287  dom cdm 4288  ran crn 4289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  dmcnvcnv  4501  rncnvcnv  4502  rncoeq  4548  cnvimass  4631  cnvimarndm  4632  dminxp  4708  cnvsn0  4732  rnsnopg  4742  dmmpt  4759  dmco  4772  cores2  4776  cnvssrndm  4785  unidmrn  4793  dfdm2  4795  cnvexg  4798  funimacnv  4918  foimacnv  5087  funcocnv2  5094  fimacnv  5239  f1opw2  5648  fopwdom  6246
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