MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnco2 Structured version   Unicode version

Theorem rnco2 5330
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 5329 . 2  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
2 df-ima 4836 . 2  |-  ( A
" ran  B )  =  ran  ( A  |`  ran  B )
31, 2eqtr4i 2434 1  |-  ran  ( A  o.  B )  =  ( A " ran  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   ran crn 4824    |` cres 4825   "cima 4826    o. ccom 4827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836
This theorem is referenced by:  dmco  5331  isf34lem7  8791  isf34lem6  8792  imasless  15154  gsumzf1o  17241  gsumzf1oOLD  17244  gsumzmhm  17280  gsumzmhmOLD  17281  gsumzinv  17292  gsumzinvOLD  17293  dprdf1o  17399  pf1rcl  18705  ovolficcss  22173  volsup  22258  uniiccdif  22279  uniioombllem3  22286  dyadmbl  22301  itg1climres  22413  cvmlift3lem6  29621  mblfinlem2  31424  volsupnfl  31431
  Copyright terms: Public domain W3C validator