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Theorem dmcoeq 4652
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 3061 . 2  |-  ( dom 
A  =  ran  B  ->  ran  B  C_  dom  A )
2 dmcosseq 4651 . 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 1285    C_ wss 2982   dom cdm 4391   ran crn 4392    o. ccom 4395
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402
This theorem is referenced by:  rncoeq  4653  dfdm2  4902  funcocnv2  5202
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