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Theorem rneqi 4562
 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 4561 . 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 1243  ran crn 4346 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356 This theorem is referenced by:  rnmpt  4582  resima  4643  resima2  4644  ima0  4684  rnuni  4735  imaundi  4736  imaundir  4737  inimass  4740  dminxp  4765  imainrect  4766  xpima1  4767  xpima2m  4768  rnresv  4780  imacnvcnv  4785  rnpropg  4800  imadmres  4813  mptpreima  4814  dmco  4829  resdif  5148  fpr  5345  fprg  5346  fliftfuns  5438  rnoprab  5587  rnmpt2  5611  qliftfuns  6190  xpassen  6304
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