Theorem cossxp 4863

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 4839	. . 3 |- Rel ( A o. B )
2		relssdmrn 4861	. . 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 4619	. . 3 |- dom ( A o. B ) C_ dom B
5		rncoss 4620	. . 3 |- ran ( A o. B ) C_ ran A
6		xpss12 4463	. . 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 416	. 2 |- ( dom ( A o. B ) X. ran ( A o. B ) ) C_ ( dom B X. ran A )
8	3, 7	sstri 3008	1 |- ( A o. B ) C_ ( dom B X. ran A )

Syntax hints:   C_ wss 2973   X. cxp 4361   dom cdm 4363   ran crn 4364   o. ccom 4367   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374

This theorem is referenced by:  coexg  4882  tposssxp  5887