Theorem cossxp2 5203

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 5202	. 2 |- ( S o. R ) C_ ( dom R X. ran S )
2		cossxp2.r	. . . 4 |- ( ph -> R C_ ( A X. B ) )
3		dmxpss2 5112	. . . 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 5113	. . . 4 |- ( S C_ ( B X. C ) -> ran S C_ C )
7	5, 6	syl 14	. . 3 |- ( ph -> ran S C_ C )
8		xpss12 4780	. . 3 |- ( ( dom R C_ A /\ ran S C_ C ) -> ( dom R X. ran S ) C_ ( A X. C ) )
9	4, 7, 8	syl2anc 411	. 2 |- ( ph -> ( dom R X. ran S ) C_ ( A X. C ) )
10	1, 9	sstrid 3203	1 |- ( ph -> ( S o. R ) C_ ( A X. C ) )

Syntax hints:   -> wi 4   C_ wss 3165   X. cxp 4671   dom cdm 4673   ran crn 4674   o. ccom 4677

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684

This theorem is referenced by: (None)