Theorem cossxp 4843

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 4819	. . 3 |- Rel ( A o. B )
2		relssdmrn 4841	. . 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 4601	. . 3 |- dom ( A o. B ) C_ dom B
5		rncoss 4602	. . 3 |- ran ( A o. B ) C_ ran A
6		xpss12 4445	. . 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 402	. 2 |- ( dom ( A o. B ) X. ran ( A o. B ) ) C_ ( dom B X. ran A )
8	3, 7	sstri 2954	1 |- ( A o. B ) C_ ( dom B X. ran A )

Syntax hints:   C_ wss 2917   X. cxp 4343   dom cdm 4345   ran crn 4346   o. ccom 4349   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356

This theorem is referenced by:  coexg  4862  tposssxp  5864