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Theorem rneqd 4852
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 4850 . 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 4624
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 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-cnv 4631  df-dm 4633  df-rn 4634
This theorem is referenced by:  resima2  4937  imaeq1  4961  imaeq2  4962  resiima  4982  elxp4  5112  elxp5  5113  funimacnv  5288  funimaexg  5296  fnima  5330  fnrnfv  5558  2ndvalg  6138  fo2nd  6153  f2ndres  6155  en1  6793  xpassen  6824  xpdom2  6825  sbthlemi4  6953  djudom  7086  exmidfodomrlemim  7194  seqeq1  10431  seqeq2  10432  seqeq3  10433  seq3val  10441  seqvalcd  10442  ennnfonelemex  12395  ennnfonelemf1  12399  restval  12639  restid2  12642  tgrest  13329  txvalex  13414  txval  13415  mopnval  13602
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