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Theorem rneqd 4858
Description: Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
Hypothesis
Ref Expression
rneqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
rneqd (𝜑 → ran 𝐴 = ran 𝐵)

Proof of Theorem rneqd
StepHypRef Expression
1 rneqd.1 . 2 (𝜑𝐴 = 𝐵)
2 rneq 4856 . 2 (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
31, 2syl 14 1 (𝜑 → ran 𝐴 = ran 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  ran crn 4629
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 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-cnv 4636  df-dm 4638  df-rn 4639
This theorem is referenced by:  resima2  4943  imaeq1  4967  imaeq2  4968  resiima  4988  elxp4  5118  elxp5  5119  funimacnv  5294  funimaexg  5302  fnima  5336  fnrnfv  5564  2ndvalg  6146  fo2nd  6161  f2ndres  6163  en1  6801  xpassen  6832  xpdom2  6833  sbthlemi4  6961  djudom  7094  exmidfodomrlemim  7202  seqeq1  10450  seqeq2  10451  seqeq3  10452  seq3val  10460  seqvalcd  10461  ennnfonelemex  12417  ennnfonelemf1  12421  restval  12699  restid2  12702  prdsex  12723  imasival  12732  tgrest  13754  txvalex  13839  txval  13840  mopnval  14027
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