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Theorem dfrn2 4466
Description: Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
Assertion
Ref Expression
dfrn2 ran A = {yx xAy}
Distinct variable group:   x,y,A

Proof of Theorem dfrn2
StepHypRef Expression
1 df-rn 4299 . 2 ran A = dom A
2 df-dm 4298 . 2 dom A = {yx yAx}
3 vex 2554 . . . . 5 y V
4 vex 2554 . . . . 5 x V
53, 4brcnv 4461 . . . 4 (yAxxAy)
65exbii 1493 . . 3 (x yAxx xAy)
76abbii 2150 . 2 {yx yAx} = {yx xAy}
81, 2, 73eqtri 2061 1 ran A = {yx xAy}
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:  dfrn3  4467  dfdm4  4470  dm0rn0  4495  dmmrnm  4497  dfrnf  4518  dfima2  4613  funcnv3  4904
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