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Theorem rneqd 4563
 Description: Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
Hypothesis
Ref Expression
rneqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
rneqd (𝜑 → ran 𝐴 = ran 𝐵)

Proof of Theorem rneqd
StepHypRef Expression
1 rneqd.1 . 2 (𝜑𝐴 = 𝐵)
2 rneq 4561 . 2 (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
31, 2syl 14 1 (𝜑 → ran 𝐴 = ran 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = 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:  resima2  4644  imaeq1  4663  imaeq2  4664  resiima  4683  elxp4  4808  elxp5  4809  funimacnv  4975  funimaexg  4983  fnima  5017  fnrnfv  5220  2ndvalg  5770  fo2nd  5785  f2ndres  5787  en1  6279  xpassen  6304  xpdom2  6305  iseqeq1  9214  iseqeq2  9215  iseqeq3  9216  iseqeq4  9217  iseqval  9220
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