Theorem cocnvss 5308

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 5307	. 2 |- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )
2		cossxp 5305	. . 3 |- ( ( S |` dom R ) o. `' ( R |` dom S ) ) C_ ( dom `' ( R |` dom S ) X. ran ( S |` dom R ) )
3		df-rn 4780	. . . . 5 |- ran ( R |` dom S ) = dom `' ( R |` dom S )
4	3	eqimss2i 3305	. . . 4 |- dom `' ( R |` dom S ) C_ ran ( R |` dom S )
5		ssid 3268	. . . 4 |- ran ( S |` dom R ) C_ ran ( S |` dom R )
6		xpss12 4877	. . . 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 430	. . 3 |- ( dom `' ( R |` dom S ) X. ran ( S |` dom R ) ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )
8	2, 7	sstri 3257	. 2 |- ( ( S |` dom R ) o. `' ( R |` dom S ) ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )
9	1, 8	eqsstri 3280	1 |- ( S o. `' R ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )

Syntax hints:   C_ wss 3220   X. cxp 4767   `'ccnv 4768   dom cdm 4769   ran crn 4770   |` cres 4771   o. ccom 4773

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781

This theorem is referenced by:  caserel  7417