Theorem cossxp2 5127

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 5126	. 2 |- ( S o. R ) C_ ( dom R X. ran S )
2		cossxp2.r	. . . 4 |- ( ph -> R C_ ( A X. B ) )
3		dmxpss2 5036	. . . 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 5037	. . . 4 |- ( S C_ ( B X. C ) -> ran S C_ C )
7	5, 6	syl 14	. . 3 |- ( ph -> ran S C_ C )
8		xpss12 4711	. . 3 |- ( ( dom R C_ A /\ ran S C_ C ) -> ( dom R X. ran S ) C_ ( A X. C ) )
9	4, 7, 8	syl2anc 409	. 2 |- ( ph -> ( dom R X. ran S ) C_ ( A X. C ) )
10	1, 9	sstrid 3153	1 |- ( ph -> ( S o. R ) C_ ( A X. C ) )

Syntax hints:   -> wi 4   C_ wss 3116   X. cxp 4602   dom cdm 4604   ran crn 4605   o. ccom 4608

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615

This theorem is referenced by: (None)