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Theorem rneqd 4857
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 4855 . 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 4628
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 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-cnv 4635  df-dm 4637  df-rn 4638
This theorem is referenced by:  resima2  4942  imaeq1  4966  imaeq2  4967  resiima  4987  elxp4  5117  elxp5  5118  funimacnv  5293  funimaexg  5301  fnima  5335  fnrnfv  5563  2ndvalg  6144  fo2nd  6159  f2ndres  6161  en1  6799  xpassen  6830  xpdom2  6831  sbthlemi4  6959  djudom  7092  exmidfodomrlemim  7200  seqeq1  10448  seqeq2  10449  seqeq3  10450  seq3val  10458  seqvalcd  10459  ennnfonelemex  12415  ennnfonelemf1  12419  restval  12694  restid2  12697  prdsex  12718  imasival  12727  tgrest  13672  txvalex  13757  txval  13758  mopnval  13945
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