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Theorem imaeq2 4847
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 4784 . . 3 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
21rneqd 4738 . 2 (𝐴 = 𝐵 → ran (𝐶𝐴) = ran (𝐶𝐵))
3 df-ima 4522 . 2 (𝐶𝐴) = ran (𝐶𝐴)
4 df-ima 4522 . 2 (𝐶𝐵) = ran (𝐶𝐵)
52, 3, 43eqtr4g 2175 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  ran crn 4510  cres 4511  cima 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  imaeq2i  4849  imaeq2d  4851  ssimaex  5450  ssimaexg  5451  isoselem  5689  f1opw2  5944  fopwdom  6698  ssenen  6713  fiintim  6785  fidcenumlemrk  6810  fidcenumlemr  6811  sbthlem2  6814  isbth  6823  ennnfonelemp1  11846  ennnfonelemnn0  11862  ctinfomlemom  11867  ctinfom  11868  tgcn  12304  iscnp4  12314  cnpnei  12315  cnima  12316  cnconst2  12329  cnrest2  12332  cnptoprest  12335  txcnp  12367  txcnmpt  12369  metcnp3  12607
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