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Theorem dmcoeq 4947
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 3231 . 2  |-  ( dom 
A  =  ran  B  ->  ran  B  C_  dom  A )
2 dmcosseq 4946 . 2  |-  ( ran 
B  C_  dom  A  ->  dom  ( A  o.  B
)  =  dom  B
)
31, 2syl 15 1  |-  ( dom 
A  =  ran  B  ->  dom  ( A  o.  B )  =  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    C_ wss 3152   dom cdm 4689   ran crn 4690    o. ccom 4693
This theorem is referenced by:  rncoeq  4948  dfdm2  5204  funcocnv2  5498
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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