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Theorem rncoss 4945
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 4944 . 2  |-  dom  ( `' B  o.  `' A )  C_  dom  `' A
2 df-rn 4700 . . 3  |-  ran  ( A  o.  B )  =  dom  `' ( A  o.  B )
3 cnvco 4865 . . . 4  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
43dmeqi 4880 . . 3  |-  dom  `' ( A  o.  B
)  =  dom  ( `' B  o.  `' A )
52, 4eqtri 2303 . 2  |-  ran  ( A  o.  B )  =  dom  ( `' B  o.  `' A )
6 df-rn 4700 . 2  |-  ran  A  =  dom  `' A
71, 5, 63sstr4i 3217 1  |-  ran  ( A  o.  B )  C_ 
ran  A
Colors of variables: wff set class
Syntax hints:    C_ wss 3152   `'ccnv 4688   dom cdm 4689   ran crn 4690    o. ccom 4693
This theorem is referenced by:  cossxp  5195  coexg  5215  fco  5398  fin23lem29  7967  fin23lem30  7968  wunco  8355  imasless  13442  gsumzf1o  15196  znleval  16508  pi1xfrcnvlem  18554  pjss1coi  22743  pj3i  22788  relexprn  24033  domrancur1b  25200  domrancur1c  25202  stoweidlem27  27776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700
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