Theorem cossxp 5192

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 5168	. . 3 |- Rel ( A o. B )
2		relssdmrn 5190	. . 3 |- ( Rel ( A o. B ) -> ( A o. B ) C_ ( dom ( A o. B ) X. ran ( A o. B ) ) )
3	1, 2	ax-mp 5	. 2 |- ( A o. B ) C_ ( dom ( A o. B ) X. ran ( A o. B ) )
4		dmcoss 4935	. . 3 |- dom ( A o. B ) C_ dom B
5		rncoss 4936	. . 3 |- ran ( A o. B ) C_ ran A
6		xpss12 4770	. . 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 426	. 2 |- ( dom ( A o. B ) X. ran ( A o. B ) ) C_ ( dom B X. ran A )
8	3, 7	sstri 3192	1 |- ( A o. B ) C_ ( dom B X. ran A )

Syntax hints:   C_ wss 3157   X. cxp 4661   dom cdm 4663   ran crn 4664   o. ccom 4667   Rel wrel 4668

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674

This theorem is referenced by:  cossxp2  5193  cocnvss  5195  coexg  5214  tposssxp  6307