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Theorem rneqd 4838
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 4836 . 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 1348  ran crn 4610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-cnv 4617  df-dm 4619  df-rn 4620
This theorem is referenced by:  resima2  4923  imaeq1  4946  imaeq2  4947  resiima  4967  elxp4  5096  elxp5  5097  funimacnv  5272  funimaexg  5280  fnima  5314  fnrnfv  5541  2ndvalg  6119  fo2nd  6134  f2ndres  6136  en1  6773  xpassen  6804  xpdom2  6805  sbthlemi4  6933  djudom  7066  exmidfodomrlemim  7165  seqeq1  10391  seqeq2  10392  seqeq3  10393  seq3val  10401  seqvalcd  10402  ennnfonelemex  12356  ennnfonelemf1  12360  restval  12572  restid2  12575  tgrest  12922  txvalex  13007  txval  13008  mopnval  13195
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