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

Theorem rncoss 5969
Description: Range of a composition. (Contributed by NM, 19-Mar-1998.)
Assertion
Ref Expression
rncoss ran (𝐴𝐵) ⊆ ran 𝐴

Proof of Theorem rncoss
StepHypRef Expression
1 dmcoss 5968 . 2 dom (𝐵𝐴) ⊆ dom 𝐴
2 df-rn 5686 . . 3 ran (𝐴𝐵) = dom (𝐴𝐵)
3 cnvco 5883 . . . 4 (𝐴𝐵) = (𝐵𝐴)
43dmeqi 5902 . . 3 dom (𝐴𝐵) = dom (𝐵𝐴)
52, 4eqtri 2760 . 2 ran (𝐴𝐵) = dom (𝐵𝐴)
6 df-rn 5686 . 2 ran 𝐴 = dom 𝐴
71, 5, 63sstr4i 4024 1 ran (𝐴𝐵) ⊆ ran 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3947  ccnv 5674  dom cdm 5675  ran crn 5676  ccom 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686
This theorem is referenced by:  cossxp  6268  fcof  6737  fcoOLD  6739  fco3OLD  6748  fin23lem29  10332  fin23lem30  10333  wunco  10724  imasless  17482  gsumzf1o  19774  znleval  21101  pi1xfrcnvlem  24563  pjss1coi  31403  pj3i  31448  smatrcl  32764  mblfinlem3  36515  mblfinlem4  36516  ismblfin  36517  relexp0a  42452  rntrclfv  42468  stoweidlem27  44729  fourierdlem42  44851  hoicvr  45250
  Copyright terms: Public domain W3C validator