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Theorem rneqd 4768
 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 4766 . 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 1331  ran crn 4540 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550 This theorem is referenced by:  resima2  4853  imaeq1  4876  imaeq2  4877  resiima  4897  elxp4  5026  elxp5  5027  funimacnv  5199  funimaexg  5207  fnima  5241  fnrnfv  5468  2ndvalg  6041  fo2nd  6056  f2ndres  6058  en1  6693  xpassen  6724  xpdom2  6725  sbthlemi4  6848  djudom  6978  exmidfodomrlemim  7057  seqeq1  10221  seqeq2  10222  seqeq3  10223  seq3val  10231  seqvalcd  10232  ennnfonelemex  11927  ennnfonelemf1  11931  restval  12126  restid2  12129  tgrest  12338  txvalex  12423  txval  12424  mopnval  12611
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