Theorem cocnvss 5254

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 5253	. 2 |- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )
2		cossxp 5251	. . 3 |- ( ( S |` dom R ) o. `' ( R |` dom S ) ) C_ ( dom `' ( R |` dom S ) X. ran ( S |` dom R ) )
3		df-rn 4730	. . . . 5 |- ran ( R |` dom S ) = dom `' ( R |` dom S )
4	3	eqimss2i 3281	. . . 4 |- dom `' ( R |` dom S ) C_ ran ( R |` dom S )
5		ssid 3244	. . . 4 |- ran ( S |` dom R ) C_ ran ( S |` dom R )
6		xpss12 4826	. . . 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 426	. . 3 |- ( dom `' ( R |` dom S ) X. ran ( S |` dom R ) ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )
8	2, 7	sstri 3233	. 2 |- ( ( S |` dom R ) o. `' ( R |` dom S ) ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )
9	1, 8	eqsstri 3256	1 |- ( S o. `' R ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )

Syntax hints:   C_ wss 3197   X. cxp 4717   `'ccnv 4718   dom cdm 4719   ran crn 4720   |` cres 4721   o. ccom 4723

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 4202  ax-pow 4258  ax-pr 4293

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 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731

This theorem is referenced by:  caserel  7254