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Theorem imaeq1 4663
 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 4606 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
21rneqd 4563 . 2 (𝐴 = 𝐵 → ran (𝐴𝐶) = ran (𝐵𝐶))
3 df-ima 4358 . 2 (𝐴𝐶) = ran (𝐴𝐶)
4 df-ima 4358 . 2 (𝐵𝐶) = ran (𝐵𝐶)
52, 3, 43eqtr4g 2097 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243  ran crn 4346   ↾ cres 4347   “ cima 4348 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by:  imaeq1i  4665  imaeq1d  4667  eceq2  6143
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