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Theorem rneqi 4587
Description: Equality inference for range. (Contributed by NM, 4-Mar-2004.)
Hypothesis
Ref Expression
rneqi.1 𝐴 = 𝐵
Assertion
Ref Expression
rneqi ran 𝐴 = ran 𝐵

Proof of Theorem rneqi
StepHypRef Expression
1 rneqi.1 . 2 𝐴 = 𝐵
2 rneq 4586 . 2 (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
31, 2ax-mp 7 1 ran 𝐴 = ran 𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1257  ran crn 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-cnv 4378  df-dm 4380  df-rn 4381
This theorem is referenced by:  rnmpt  4607  resima  4668  resima2  4669  ima0  4709  rnuni  4760  imaundi  4761  imaundir  4762  inimass  4765  dminxp  4790  imainrect  4791  xpima1  4792  xpima2m  4793  rnresv  4805  imacnvcnv  4810  rnpropg  4825  imadmres  4838  mptpreima  4839  dmco  4854  resdif  5173  fpr  5370  fprg  5371  fliftfuns  5463  rnoprab  5612  rnmpt2  5636  qliftfuns  6218  xpassen  6332
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