Theorem cocnvss 5146

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 5145	. 2 |- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )
2		cossxp 5143	. . 3 |- ( ( S |` dom R ) o. `' ( R |` dom S ) ) C_ ( dom `' ( R |` dom S ) X. ran ( S |` dom R ) )
3		df-rn 4631	. . . . 5 |- ran ( R |` dom S ) = dom `' ( R |` dom S )
4	3	eqimss2i 3210	. . . 4 |- dom `' ( R |` dom S ) C_ ran ( R |` dom S )
5		ssid 3173	. . . 4 |- ran ( S |` dom R ) C_ ran ( S |` dom R )
6		xpss12 4727	. . . 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 3162	. 2 |- ( ( S |` dom R ) o. `' ( R |` dom S ) ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )
9	1, 8	eqsstri 3185	1 |- ( S o. `' R ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )

Syntax hints:   C_ wss 3127   X. cxp 4618   `'ccnv 4619   dom cdm 4620   ran crn 4621   |` cres 4622   o. ccom 4624

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632

This theorem is referenced by:  caserel  7076