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Theorem rneqd 4856
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 4854 . 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 1353   ran crn 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-cnv 4634  df-dm 4636  df-rn 4637
This theorem is referenced by:  resima2  4941  imaeq1  4965  imaeq2  4966  resiima  4986  elxp4  5116  elxp5  5117  funimacnv  5292  funimaexg  5300  fnima  5334  fnrnfv  5562  2ndvalg  6143  fo2nd  6158  f2ndres  6160  en1  6798  xpassen  6829  xpdom2  6830  sbthlemi4  6958  djudom  7091  exmidfodomrlemim  7199  seqeq1  10447  seqeq2  10448  seqeq3  10449  seq3val  10457  seqvalcd  10458  ennnfonelemex  12414  ennnfonelemf1  12418  restval  12693  restid2  12696  prdsex  12717  imasival  12726  tgrest  13639  txvalex  13724  txval  13725  mopnval  13912
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