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Theorem cocnvss 5068
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
StepHypRef Expression
1 cocnvres 5067 . 2  |-  ( S  o.  `' R )  =  ( ( S  |`  dom  R )  o.  `' ( R  |`  dom  S ) )
2 cossxp 5065 . . 3  |-  ( ( S  |`  dom  R )  o.  `' ( R  |`  dom  S ) ) 
C_  ( dom  `' ( R  |`  dom  S
)  X.  ran  ( S  |`  dom  R ) )
3 df-rn 4554 . . . . 5  |-  ran  ( R  |`  dom  S )  =  dom  `' ( R  |`  dom  S )
43eqimss2i 3155 . . . 4  |-  dom  `' ( R  |`  dom  S
)  C_  ran  ( R  |`  dom  S )
5 ssid 3118 . . . 4  |-  ran  ( S  |`  dom  R ) 
C_  ran  ( S  |` 
dom  R )
6 xpss12 4650 . . . 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 ) ) )
74, 5, 6mp2an 423 . . 3  |-  ( dom  `' ( R  |`  dom  S )  X.  ran  ( S  |`  dom  R
) )  C_  ( ran  ( R  |`  dom  S
)  X.  ran  ( S  |`  dom  R ) )
82, 7sstri 3107 . 2  |-  ( ( S  |`  dom  R )  o.  `' ( R  |`  dom  S ) ) 
C_  ( ran  ( R  |`  dom  S )  X.  ran  ( S  |`  dom  R ) )
91, 8eqsstri 3130 1  |-  ( S  o.  `' R ) 
C_  ( ran  ( R  |`  dom  S )  X.  ran  ( S  |`  dom  R ) )
Colors of variables: wff set class
Syntax hints:    C_ wss 3072    X. cxp 4541   `'ccnv 4542   dom cdm 4543   ran crn 4544    |` cres 4545    o. ccom 4547
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555
This theorem is referenced by:  caserel  6976
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