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Theorem imaeq1 4713
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 4654 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
21rneqd 4611 . 2 (𝐴 = 𝐵 → ran (𝐴𝐶) = ran (𝐵𝐶))
3 df-ima 4404 . 2 (𝐴𝐶) = ran (𝐴𝐶)
4 df-ima 4404 . 2 (𝐵𝐶) = ran (𝐵𝐶)
52, 3, 43eqtr4g 2140 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  ran crn 4392  cres 4393  cima 4394
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 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404
This theorem is referenced by:  imaeq1i  4715  imaeq1d  4717  eceq2  6230
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