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Theorem rncoss 5294
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 5293 . 2 dom (𝐵𝐴) ⊆ dom 𝐴
2 df-rn 5039 . . 3 ran (𝐴𝐵) = dom (𝐴𝐵)
3 cnvco 5218 . . . 4 (𝐴𝐵) = (𝐵𝐴)
43dmeqi 5234 . . 3 dom (𝐴𝐵) = dom (𝐵𝐴)
52, 4eqtri 2631 . 2 ran (𝐴𝐵) = dom (𝐵𝐴)
6 df-rn 5039 . 2 ran 𝐴 = dom 𝐴
71, 5, 63sstr4i 3606 1 ran (𝐴𝐵) ⊆ ran 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3539  ccnv 5027  dom cdm 5028  ran crn 5029  ccom 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039
This theorem is referenced by:  cossxp  5561  fco  5957  fin23lem29  9023  fin23lem30  9024  wunco  9411  imasless  15969  gsumzf1o  18082  znleval  19667  pi1xfrcnvlem  22595  pjss1coi  28212  pj3i  28257  smatrcl  28996  mblfinlem3  32421  mblfinlem4  32422  ismblfin  32423  relexp0a  36830  rntrclfv  36846  fco3  38219  stoweidlem27  38724  fourierdlem42  38846  hoicvr  39242
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