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Theorem rnco2 4828
 Description: The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
Assertion
Ref Expression
rnco2 ran (𝐴𝐵) = (𝐴 “ ran 𝐵)

Proof of Theorem rnco2
StepHypRef Expression
1 rnco 4827 . 2 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
2 df-ima 4358 . 2 (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵)
31, 2eqtr4i 2063 1 ran (𝐴𝐵) = (𝐴 “ ran 𝐵)
 Colors of variables: wff set class Syntax hints:   = wceq 1243  ran crn 4346   ↾ cres 4347   “ cima 4348   ∘ ccom 4349 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-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by:  dmco  4829
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