Theorem dmcoeq 4952

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 3248	. 2 |- ( dom A = ran B -> ran B C_ dom A )
2		dmcosseq 4951	. 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 1373   C_ wss 3166   dom cdm 4676   ran crn 4677   o. ccom 4680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687

This theorem is referenced by:  rncoeq  4953  dfdm2  5218  funcocnv2  5549