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Theorem rneqd 4632
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 4630 . 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 1287   ran crn 4412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-cnv 4419  df-dm 4421  df-rn 4422
This theorem is referenced by:  resima2  4713  imaeq1  4736  imaeq2  4737  resiima  4757  elxp4  4884  elxp5  4885  funimacnv  5055  funimaexg  5063  fnima  5097  fnrnfv  5314  2ndvalg  5871  fo2nd  5886  f2ndres  5888  en1  6468  xpassen  6498  xpdom2  6499  sbthlemi4  6613  djudom  6731  exmidfodomrlemim  6771  iseqeq1  9782  iseqeq2  9783  iseqeq3  9784  iseqeq4  9785  iseqval  9788  iseqvalt  9790
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