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Theorem rneqi 4663
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 4662 . 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 1289  ran crn 4439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-cnv 4446  df-dm 4448  df-rn 4449
This theorem is referenced by:  rnmpt  4683  resima  4745  resima2  4746  ima0  4791  rnuni  4843  imaundi  4844  imaundir  4845  inimass  4848  dminxp  4875  imainrect  4876  xpima1  4877  xpima2m  4878  rnresv  4890  imacnvcnv  4895  rnpropg  4910  imadmres  4923  mptpreima  4924  dmco  4939  resdif  5275  fpr  5479  fprg  5480  fliftfuns  5577  rnoprab  5731  rnmpt2  5755  qliftfuns  6376  xpassen  6546  sbthlemi6  6671
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