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Theorem rnco2 5680
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 5679 . 2 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
2 df-ima 5156 . 2 (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵)
31, 2eqtr4i 2676 1 ran (𝐴𝐵) = (𝐴 “ ran 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  ran crn 5144  cres 5145  cima 5146  ccom 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  dmco  5681  isf34lem7  9239  isf34lem6  9240  imasless  16247  gsumzf1o  18359  gsumzmhm  18383  gsumzinv  18391  dprdf1o  18477  pf1rcl  19761  ovolficcss  23284  volsup  23370  uniiccdif  23392  uniioombllem3  23399  dyadmbl  23414  itg1climres  23526  cvmlift3lem6  31432  mblfinlem2  33577  volsupnfl  33584
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