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Theorem rncoss 5418
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 5417 . 2 dom (𝐵𝐴) ⊆ dom 𝐴
2 df-rn 5154 . . 3 ran (𝐴𝐵) = dom (𝐴𝐵)
3 cnvco 5340 . . . 4 (𝐴𝐵) = (𝐵𝐴)
43dmeqi 5357 . . 3 dom (𝐴𝐵) = dom (𝐵𝐴)
52, 4eqtri 2673 . 2 ran (𝐴𝐵) = dom (𝐵𝐴)
6 df-rn 5154 . 2 ran 𝐴 = dom 𝐴
71, 5, 63sstr4i 3677 1 ran (𝐴𝐵) ⊆ ran 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3607  ccnv 5142  dom cdm 5143  ran crn 5144  ccom 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154
This theorem is referenced by:  cossxp  5696  fco  6096  fin23lem29  9201  fin23lem30  9202  wunco  9593  imasless  16247  gsumzf1o  18359  znleval  19951  pi1xfrcnvlem  22902  pjss1coi  29150  pj3i  29195  smatrcl  29990  mblfinlem3  33578  mblfinlem4  33579  ismblfin  33580  relexp0a  38325  rntrclfv  38341  fco3  39735  stoweidlem27  40562  fourierdlem42  40684  hoicvr  41083
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