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Theorem dmcoeq 5167
Description: Domain of a composition. (Contributed by NM, 19-Mar-1998.)
Assertion
Ref Expression
dmcoeq  |-  ( dom 
A  =  ran  B  ->  dom  ( A  o.  B )  =  dom  B )

Proof of Theorem dmcoeq
StepHypRef Expression
1 eqimss2 3387 . 2  |-  ( dom 
A  =  ran  B  ->  ran  B  C_  dom  A )
2 dmcosseq 5166 . 2  |-  ( ran 
B  C_  dom  A  ->  dom  ( A  o.  B
)  =  dom  B
)
31, 2syl 16 1  |-  ( dom 
A  =  ran  B  ->  dom  ( A  o.  B )  =  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    C_ wss 3306   dom cdm 4907   ran crn 4908    o. ccom 4911
This theorem is referenced by:  rncoeq  5168  dfdm2  5430  funcocnv2  5729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918
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