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Theorem rncoss 4961
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 4960 . 2  |-  dom  ( `' B  o.  `' A )  C_  dom  `' A
2 df-rn 4716 . . 3  |-  ran  ( A  o.  B )  =  dom  `' ( A  o.  B )
3 cnvco 4881 . . . 4  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
43dmeqi 4896 . . 3  |-  dom  `' ( A  o.  B
)  =  dom  ( `' B  o.  `' A )
52, 4eqtri 2316 . 2  |-  ran  ( A  o.  B )  =  dom  ( `' B  o.  `' A )
6 df-rn 4716 . 2  |-  ran  A  =  dom  `' A
71, 5, 63sstr4i 3230 1  |-  ran  ( A  o.  B )  C_ 
ran  A
Colors of variables: wff set class
Syntax hints:    C_ wss 3165   `'ccnv 4704   dom cdm 4705   ran crn 4706    o. ccom 4709
This theorem is referenced by:  cossxp  5211  coexg  5231  fco  5414  fin23lem29  7983  fin23lem30  7984  wunco  8371  imasless  13458  gsumzf1o  15212  znleval  16524  pi1xfrcnvlem  18570  pjss1coi  22759  pj3i  22804  relexprn  24048  domrancur1b  25303  domrancur1c  25305  stoweidlem27  27879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716
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