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Theorem dmcoeq 4693
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 3077 . 2  |-  ( dom 
A  =  ran  B  ->  ran  B  C_  dom  A )
2 dmcosseq 4692 . 2  |-  ( ran 
B  C_  dom  A  ->  dom  ( A  o.  B
)  =  dom  B
)
31, 2syl 14 1  |-  ( dom 
A  =  ran  B  ->  dom  ( A  o.  B )  =  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    C_ wss 2997   dom cdm 4428   ran crn 4429    o. ccom 4432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439
This theorem is referenced by:  rncoeq  4694  dfdm2  4952  funcocnv2  5262
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