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Theorem cocnvss 5034
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 5033 . 2  |-  ( S  o.  `' R )  =  ( ( S  |`  dom  R )  o.  `' ( R  |`  dom  S ) )
2 cossxp 5031 . . 3  |-  ( ( S  |`  dom  R )  o.  `' ( R  |`  dom  S ) ) 
C_  ( dom  `' ( R  |`  dom  S
)  X.  ran  ( S  |`  dom  R ) )
3 df-rn 4520 . . . . 5  |-  ran  ( R  |`  dom  S )  =  dom  `' ( R  |`  dom  S )
43eqimss2i 3124 . . . 4  |-  dom  `' ( R  |`  dom  S
)  C_  ran  ( R  |`  dom  S )
5 ssid 3087 . . . 4  |-  ran  ( S  |`  dom  R ) 
C_  ran  ( S  |` 
dom  R )
6 xpss12 4616 . . . 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 422 . . 3  |-  ( dom  `' ( R  |`  dom  S )  X.  ran  ( S  |`  dom  R
) )  C_  ( ran  ( R  |`  dom  S
)  X.  ran  ( S  |`  dom  R ) )
82, 7sstri 3076 . 2  |-  ( ( S  |`  dom  R )  o.  `' ( R  |`  dom  S ) ) 
C_  ( ran  ( R  |`  dom  S )  X.  ran  ( S  |`  dom  R ) )
91, 8eqsstri 3099 1  |-  ( S  o.  `' R ) 
C_  ( ran  ( R  |`  dom  S )  X.  ran  ( S  |`  dom  R ) )
Colors of variables: wff set class
Syntax hints:    C_ wss 3041    X. cxp 4507   `'ccnv 4508   dom cdm 4509   ran crn 4510    |` cres 4511    o. ccom 4513
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521
This theorem is referenced by:  caserel  6940
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