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Theorem rnco2 5640
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 5639 . 2 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
2 df-ima 5125 . 2 (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵)
31, 2eqtr4i 2646 1 ran (𝐴𝐵) = (𝐴 “ ran 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1482  ran crn 5113  cres 5114  cima 5115  ccom 5116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125
This theorem is referenced by:  dmco  5641  isf34lem7  9198  isf34lem6  9199  imasless  16194  gsumzf1o  18307  gsumzmhm  18331  gsumzinv  18339  dprdf1o  18425  pf1rcl  19707  ovolficcss  23232  volsup  23318  uniiccdif  23340  uniioombllem3  23347  dyadmbl  23362  itg1climres  23475  cvmlift3lem6  31291  mblfinlem2  33427  volsupnfl  33434
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