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Theorem rneq 4561
 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 4509 . . 3 (𝐴 = 𝐵𝐴 = 𝐵)
21dmeqd 4537 . 2 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
3 df-rn 4356 . 2 ran 𝐴 = dom 𝐴
4 df-rn 4356 . 2 ran 𝐵 = dom 𝐵
52, 3, 43eqtr4g 2097 1 (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  ◡ccnv 4344  dom cdm 4345  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:  rneqi  4562  rneqd  4563  xpima1  4767  feq1  5030  foeq1  5102
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