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Theorem imaeq1 5917
Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
imaeq1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem imaeq1
StepHypRef Expression
1 reseq1 5840 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
21rneqd 5801 . 2 (𝐴 = 𝐵 → ran (𝐴𝐶) = ran (𝐵𝐶))
3 df-ima 5561 . 2 (𝐴𝐶) = ran (𝐴𝐶)
4 df-ima 5561 . 2 (𝐵𝐶) = ran (𝐵𝐶)
52, 3, 43eqtr4g 2878 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  ran crn 5549  cres 5550  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  imaeq1i  5919  imaeq1d  5921  suppval  7821  eceq2  8318  marypha1lem  8885  marypha1  8886  ackbij2lem2  9650  ackbij2lem3  9651  r1om  9654  limsupval  14819  isacs1i  16916  mreacs  16917  islindf  20884  iscnp  21773  xkoccn  22155  xkohaus  22189  xkoco1cn  22193  xkoco2cn  22194  xkococnlem  22195  xkococn  22196  xkoinjcn  22223  fmval  22479  fmf  22481  utoptop  22770  restutop  22773  restutopopn  22774  ustuqtoplem  22775  ustuqtop1  22777  ustuqtop2  22778  ustuqtop4  22780  ustuqtop5  22781  utopsnneiplem  22783  utopsnnei  22785  neipcfilu  22832  psmetutop  23104  cfilfval  23794  elply2  24713  coeeu  24742  coelem  24743  coeeq  24744  dmarea  25462  mclsax  32713  tailfval  33617  bj-cleq  34171  bj-funun  34426  poimirlem15  34788  poimirlem24  34797  brtrclfv2  39950  liminfval  41916  isomgreqve  43867  isomgrsym  43878  isomgrtr  43881  ushrisomgr  43883
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