Theorem cossxp 4966

Description: Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)

Assertion

Ref	Expression
cossxp	|- ( A o. B ) C_ ( dom B X. ran A )

Proof of Theorem cossxp

Step	Hyp	Ref	Expression
1		relco 4942	. . 3 |- Rel ( A o. B )
2		relssdmrn 4964	. . 3 |- ( Rel ( A o. B ) -> ( A o. B ) C_ ( dom ( A o. B ) X. ran ( A o. B ) ) )
3	1, 2	ax-mp 7	. 2 |- ( A o. B ) C_ ( dom ( A o. B ) X. ran ( A o. B ) )
4		dmcoss 4715	. . 3 |- dom ( A o. B ) C_ dom B
5		rncoss 4716	. . 3 |- ran ( A o. B ) C_ ran A
6		xpss12 4558	. . 3 |- ( ( dom ( A o. B ) C_ dom B /\ ran ( A o. B ) C_ ran A ) -> ( dom ( A o. B ) X. ran ( A o. B ) ) C_ ( dom B X. ran A ) )
7	4, 5, 6	mp2an 418	. 2 |- ( dom ( A o. B ) X. ran ( A o. B ) ) C_ ( dom B X. ran A )
8	3, 7	sstri 3035	1 |- ( A o. B ) C_ ( dom B X. ran A )

Syntax hints:   C_ wss 3000   X. cxp 4450   dom cdm 4452   ran crn 4453   o. ccom 4456   Rel wrel 4457

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463

This theorem is referenced by:  cossxp2  4967  cocnvss  4969  coexg  4988  tposssxp  6028