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

Theorem rnco2 6206
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 6205 . 2 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
2 df-ima 5647 . 2 (𝐴 “ ran 𝐵) = ran (𝐴 ↾ ran 𝐵)
31, 2eqtr4i 2764 1 ran (𝐴𝐵) = (𝐴 “ ran 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  ran crn 5635  cres 5636  cima 5637  ccom 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647
This theorem is referenced by:  dmco  6207  isf34lem7  10320  isf34lem6  10321  imasless  17427  gsumzf1o  19694  gsumzmhm  19719  gsumzinv  19727  dprdf1o  19816  pf1rcl  21731  ovolficcss  24849  volsup  24936  uniiccdif  24958  uniioombllem3  24965  dyadmbl  24980  itg1climres  25095  cvmlift3lem6  33975  mblfinlem2  36162  volsupnfl  36169
  Copyright terms: Public domain W3C validator