Theorem dmcoeq 4876

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

Step	Hyp	Ref	Expression
1		eqimss2 3197	. 2 |- ( dom A = ran B -> ran B C_ dom A )
2		dmcosseq 4875	. 2 |- ( ran B C_ dom A -> dom ( A o. B ) = dom B )
3	1, 2	syl 14	1 |- ( dom A = ran B -> dom ( A o. B ) = dom B )

Syntax hints:   -> wi 4   = wceq 1343   C_ wss 3116   dom cdm 4604   ran crn 4605   o. ccom 4608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615

This theorem is referenced by:  rncoeq  4877  dfdm2  5138  funcocnv2  5457