Theorem cossxp2 5062

Description: The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.)

Hypotheses

Ref	Expression
cossxp2.r	|- ( ph -> R C_ ( A X. B ) )
cossxp2.s	|- ( ph -> S C_ ( B X. C ) )

Assertion

Ref	Expression
cossxp2	|- ( ph -> ( S o. R ) C_ ( A X. C ) )

Proof of Theorem cossxp2

Step	Hyp	Ref	Expression
1		cossxp 5061	. 2 |- ( S o. R ) C_ ( dom R X. ran S )
2		cossxp2.r	. . . 4 |- ( ph -> R C_ ( A X. B ) )
3		dmxpss2 4971	. . . 4 |- ( R C_ ( A X. B ) -> dom R C_ A )
4	2, 3	syl 14	. . 3 |- ( ph -> dom R C_ A )
5		cossxp2.s	. . . 4 |- ( ph -> S C_ ( B X. C ) )
6		rnxpss2 4972	. . . 4 |- ( S C_ ( B X. C ) -> ran S C_ C )
7	5, 6	syl 14	. . 3 |- ( ph -> ran S C_ C )
8		xpss12 4646	. . 3 |- ( ( dom R C_ A /\ ran S C_ C ) -> ( dom R X. ran S ) C_ ( A X. C ) )
9	4, 7, 8	syl2anc 408	. 2 |- ( ph -> ( dom R X. ran S ) C_ ( A X. C ) )
10	1, 9	sstrid 3108	1 |- ( ph -> ( S o. R ) C_ ( A X. C ) )

Syntax hints:   -> wi 4   C_ wss 3071   X. cxp 4537   dom cdm 4539   ran crn 4540   o. ccom 4543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550

This theorem is referenced by: (None)