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Theorem rneqd 4775
Description: Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
Hypothesis
Ref Expression
rneqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
rneqd  |-  ( ph  ->  ran  A  =  ran  B )

Proof of Theorem rneqd
StepHypRef Expression
1 rneqd.1 . 2  |-  ( ph  ->  A  =  B )
2 rneq 4773 . 2  |-  ( A  =  B  ->  ran  A  =  ran  B )
31, 2syl 14 1  |-  ( ph  ->  ran  A  =  ran  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332   ran crn 4547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-cnv 4554  df-dm 4556  df-rn 4557
This theorem is referenced by:  resima2  4860  imaeq1  4883  imaeq2  4884  resiima  4904  elxp4  5033  elxp5  5034  funimacnv  5206  funimaexg  5214  fnima  5248  fnrnfv  5475  2ndvalg  6048  fo2nd  6063  f2ndres  6065  en1  6700  xpassen  6731  xpdom2  6732  sbthlemi4  6855  djudom  6985  exmidfodomrlemim  7073  seqeq1  10251  seqeq2  10252  seqeq3  10253  seq3val  10261  seqvalcd  10262  ennnfonelemex  11961  ennnfonelemf1  11965  restval  12163  restid2  12166  tgrest  12375  txvalex  12460  txval  12461  mopnval  12648
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