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Theorem rneq 4869
Description: Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
rneq (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)

Proof of Theorem rneq
StepHypRef Expression
1 cnveq 4816 . . 3 (𝐴 = 𝐵𝐴 = 𝐵)
21dmeqd 4844 . 2 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
3 df-rn 4652 . 2 ran 𝐴 = dom 𝐴
4 df-rn 4652 . 2 ran 𝐵 = dom 𝐵
52, 3, 43eqtr4g 2247 1 (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  ccnv 4640  dom cdm 4641  ran crn 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-cnv 4649  df-dm 4651  df-rn 4652
This theorem is referenced by:  rneqi  4870  rneqd  4871  xpima1  5090  feq1  5363  foeq1  5449  ixpsnf1o  6754  imasex  12748
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