Metamath Proof Explorer

Theorem dmcoeq

Description: Domain of a composition. (Contributed by NM, 19-Mar-1998)

Ref Expression
Assertion dmcoeq 
|- ( dom A = ran B -> dom ( A o. B ) = dom B )

Proof

Step Hyp Ref Expression
1 eqimss2 
 |-  ( dom A = ran B -> ran B C_ dom A )
2 dmcosseq 
 |-  ( ran B C_ dom A -> dom ( A o. B ) = dom B )
3 1 2 syl 
 |-  ( dom A = ran B -> dom ( A o. B ) = dom B )