Theorem cocnvss 5123

Description: Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.)

Assertion

Ref	Expression
cocnvss	|- ( S o. `' R ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )

Proof of Theorem cocnvss

Step	Hyp	Ref	Expression
1		cocnvres 5122	. 2 |- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )
2		cossxp 5120	. . 3 |- ( ( S |` dom R ) o. `' ( R |` dom S ) ) C_ ( dom `' ( R |` dom S ) X. ran ( S |` dom R ) )
3		df-rn 4609	. . . . 5 |- ran ( R |` dom S ) = dom `' ( R |` dom S )
4	3	eqimss2i 3194	. . . 4 |- dom `' ( R |` dom S ) C_ ran ( R |` dom S )
5		ssid 3157	. . . 4 |- ran ( S |` dom R ) C_ ran ( S |` dom R )
6		xpss12 4705	. . . 4 |- ( ( dom `' ( R |` dom S ) C_ ran ( R |` dom S ) /\ ran ( S |` dom R ) C_ ran ( S |` dom R ) ) -> ( dom `' ( R |` dom S ) X. ran ( S |` dom R ) ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) ) )
7	4, 5, 6	mp2an 423	. . 3 |- ( dom `' ( R |` dom S ) X. ran ( S |` dom R ) ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )
8	2, 7	sstri 3146	. 2 |- ( ( S |` dom R ) o. `' ( R |` dom S ) ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )
9	1, 8	eqsstri 3169	1 |- ( S o. `' R ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )

Syntax hints:   C_ wss 3111   X. cxp 4596   `'ccnv 4597   dom cdm 4598   ran crn 4599   |` cres 4600   o. ccom 4602

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610

This theorem is referenced by:  caserel  7043