Theorem cocnvss 4969

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 4968	. 2 |- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )
2		cossxp 4966	. . 3 |- ( ( S |` dom R ) o. `' ( R |` dom S ) ) C_ ( dom `' ( R |` dom S ) X. ran ( S |` dom R ) )
3		df-rn 4463	. . . . 5 |- ran ( R |` dom S ) = dom `' ( R |` dom S )
4	3	eqimss2i 3082	. . . 4 |- dom `' ( R |` dom S ) C_ ran ( R |` dom S )
5		ssid 3045	. . . 4 |- ran ( S |` dom R ) C_ ran ( S |` dom R )
6		xpss12 4558	. . . 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 418	. . 3 |- ( dom `' ( R |` dom S ) X. ran ( S |` dom R ) ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )
8	2, 7	sstri 3035	. 2 |- ( ( S |` dom R ) o. `' ( R |` dom S ) ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )
9	1, 8	eqsstri 3057	1 |- ( S o. `' R ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )

Syntax hints:   C_ wss 3000   X. cxp 4450   `'ccnv 4451   dom cdm 4452   ran crn 4453   |` cres 4454   o. ccom 4456

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464

This theorem is referenced by:  caserel  6832