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Theorem rneqd 4590
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 4588 . 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 1259  ran crn 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-cnv 4380  df-dm 4382  df-rn 4383
This theorem is referenced by:  resima2  4671  imaeq1  4690  imaeq2  4691  resiima  4710  elxp4  4835  elxp5  4836  funimacnv  5002  funimaexg  5010  fnima  5044  fnrnfv  5247  2ndvalg  5797  fo2nd  5812  f2ndres  5814  en1  6309  xpassen  6334  xpdom2  6335  iseqeq1  9377  iseqeq2  9378  iseqeq3  9379  iseqeq4  9380  iseqval  9383
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