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Theorem imaeq2 4966
Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
imaeq2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem imaeq2
StepHypRef Expression
1 reseq2 4902 . . 3 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
21rneqd 4856 . 2 (𝐴 = 𝐵 → ran (𝐶𝐴) = ran (𝐶𝐵))
3 df-ima 4639 . 2 (𝐶𝐴) = ran (𝐶𝐴)
4 df-ima 4639 . 2 (𝐶𝐵) = ran (𝐶𝐵)
52, 3, 43eqtr4g 2235 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  ran crn 4627  cres 4628  cima 4629
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639
This theorem is referenced by:  imaeq2i  4968  imaeq2d  4970  ssimaex  5577  ssimaexg  5578  isoselem  5820  f1opw2  6076  fopwdom  6835  ssenen  6850  fiintim  6927  fidcenumlemrk  6952  fidcenumlemr  6953  sbthlem2  6956  isbth  6965  ennnfonelemp1  12406  ennnfonelemnn0  12422  ctinfomlemom  12427  ctinfom  12428  tgcn  13678  iscnp4  13688  cnpnei  13689  cnima  13690  cnconst2  13703  cnrest2  13706  cnptoprest  13709  txcnp  13741  txcnmpt  13743  metcnp3  13981
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