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Theorem imaeq2 4688
 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 4629 . . 3 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
21rneqd 4585 . 2 (𝐴 = 𝐵 → ran (𝐶𝐴) = ran (𝐶𝐵))
3 df-ima 4378 . 2 (𝐶𝐴) = ran (𝐶𝐴)
4 df-ima 4378 . 2 (𝐶𝐵) = ran (𝐶𝐵)
52, 3, 43eqtr4g 2139 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1285  ran crn 4366   ↾ cres 4367   “ cima 4368 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-cnv 4373  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378 This theorem is referenced by:  imaeq2i  4690  imaeq2d  4692  ssimaex  5260  ssimaexg  5261  isoselem  5484  f1opw2  5731  fopwdom  6370
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