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Theorem rnco2 5507
Description: The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
Assertion
Ref Expression
rnco2  |-  ran  ( A  o.  B )  =  ( A " ran  B )

Proof of Theorem rnco2
StepHypRef Expression
1 rnco 5506 . 2  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
2 df-ima 5007 . 2  |-  ( A
" ran  B )  =  ran  ( A  |`  ran  B )
31, 2eqtr4i 2494 1  |-  ran  ( A  o.  B )  =  ( A " ran  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374   ran crn 4995    |` cres 4996   "cima 4997    o. ccom 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007
This theorem is referenced by:  dmco  5508  isf34lem7  8750  isf34lem6  8751  imasless  14786  gsumzf1o  16703  gsumzf1oOLD  16706  gsumzmhm  16743  gsumzmhmOLD  16744  gsumzinv  16755  gsumzinvOLD  16756  dprdf1o  16864  pf1rcl  18151  ovolficcss  21611  volsup  21696  uniiccdif  21717  uniioombllem3  21724  dyadmbl  21739  itg1climres  21851  cvmlift3lem6  28397  mblfinlem2  29618  volsupnfl  29625
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