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Theorem rncoss 5138
Description: Range of a composition. (Contributed by NM, 19-Mar-1998.)
Assertion
Ref Expression
rncoss  |-  ran  ( A  o.  B )  C_ 
ran  A

Proof of Theorem rncoss
StepHypRef Expression
1 dmcoss 5137 . 2  |-  dom  ( `' B  o.  `' A )  C_  dom  `' A
2 df-rn 4891 . . 3  |-  ran  ( A  o.  B )  =  dom  `' ( A  o.  B )
3 cnvco 5058 . . . 4  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
43dmeqi 5073 . . 3  |-  dom  `' ( A  o.  B
)  =  dom  ( `' B  o.  `' A )
52, 4eqtri 2458 . 2  |-  ran  ( A  o.  B )  =  dom  ( `' B  o.  `' A )
6 df-rn 4891 . 2  |-  ran  A  =  dom  `' A
71, 5, 63sstr4i 3389 1  |-  ran  ( A  o.  B )  C_ 
ran  A
Colors of variables: wff set class
Syntax hints:    C_ wss 3322   `'ccnv 4879   dom cdm 4880   ran crn 4881    o. ccom 4884
This theorem is referenced by:  cossxp  5394  fco  5602  fin23lem29  8223  fin23lem30  8224  wunco  8610  imasless  13767  gsumzf1o  15521  znleval  16837  pi1xfrcnvlem  19083  pjss1coi  23668  pj3i  23713  relexprn  25138  mblfinlem3  26247  mblfinlem4  26248  ismblfin  26249  stoweidlem27  27754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891
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