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

Theorem rncoss 5808
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 5807 . 2 dom (𝐵𝐴) ⊆ dom 𝐴
2 df-rn 5530 . . 3 ran (𝐴𝐵) = dom (𝐴𝐵)
3 cnvco 5720 . . . 4 (𝐴𝐵) = (𝐵𝐴)
43dmeqi 5737 . . 3 dom (𝐴𝐵) = dom (𝐵𝐴)
52, 4eqtri 2821 . 2 ran (𝐴𝐵) = dom (𝐵𝐴)
6 df-rn 5530 . 2 ran 𝐴 = dom 𝐴
71, 5, 63sstr4i 3958 1 ran (𝐴𝐵) ⊆ ran 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3881  ccnv 5518  dom cdm 5519  ran crn 5520  ccom 5523
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530
This theorem is referenced by:  cossxp  6091  fco  6505  fin23lem29  9752  fin23lem30  9753  wunco  10144  imasless  16805  gsumzf1o  19025  znleval  20246  pi1xfrcnvlem  23661  pjss1coi  29946  pj3i  29991  smatrcl  31149  mblfinlem3  35096  mblfinlem4  35097  ismblfin  35098  relexp0a  40415  rntrclfv  40431  fco3  41855  stoweidlem27  42667  fourierdlem42  42789  hoicvr  43185
  Copyright terms: Public domain W3C validator