Theorem cossxp2 5251

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 5250	. 2 |- ( S o. R ) C_ ( dom R X. ran S )
2		cossxp2.r	. . . 4 |- ( ph -> R C_ ( A X. B ) )
3		dmxpss2 5160	. . . 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 5161	. . . 4 |- ( S C_ ( B X. C ) -> ran S C_ C )
7	5, 6	syl 14	. . 3 |- ( ph -> ran S C_ C )
8		xpss12 4825	. . 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 3235	1 |- ( ph -> ( S o. R ) C_ ( A X. C ) )

Syntax hints:   -> wi 4   C_ wss 3197   X. cxp 4716   dom cdm 4718   ran crn 4719   o. ccom 4722

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729

This theorem is referenced by: (None)