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Theorem cocnvres 5058
Description: Restricting a relation and a converse relation when they are composed together (Contributed by BJ, 10-Jul-2022.)
Assertion
Ref Expression
cocnvres  |-  ( S  o.  `' R )  =  ( ( S  |`  dom  R )  o.  `' ( R  |`  dom  S ) )

Proof of Theorem cocnvres
StepHypRef Expression
1 resss 4838 . . . 4  |-  ( S  |`  dom  R )  C_  S
2 dmss 4733 . . . 4  |-  ( ( S  |`  dom  R ) 
C_  S  ->  dom  ( S  |`  dom  R
)  C_  dom  S )
31, 2ax-mp 5 . . 3  |-  dom  ( S  |`  dom  R ) 
C_  dom  S
4 cores2 5046 . . 3  |-  ( dom  ( S  |`  dom  R
)  C_  dom  S  -> 
( ( S  |`  dom  R )  o.  `' ( `' `' R  |`  dom  S
) )  =  ( ( S  |`  dom  R
)  o.  `' R
) )
53, 4ax-mp 5 . 2  |-  ( ( S  |`  dom  R )  o.  `' ( `' `' R  |`  dom  S
) )  =  ( ( S  |`  dom  R
)  o.  `' R
)
6 rescnvcnv 4996 . . . 4  |-  ( `' `' R  |`  dom  S
)  =  ( R  |`  dom  S )
76cnveqi 4709 . . 3  |-  `' ( `' `' R  |`  dom  S
)  =  `' ( R  |`  dom  S )
87coeq2i 4694 . 2  |-  ( ( S  |`  dom  R )  o.  `' ( `' `' R  |`  dom  S
) )  =  ( ( S  |`  dom  R
)  o.  `' ( R  |`  dom  S ) )
9 dfdm4 4726 . . . 4  |-  dom  R  =  ran  `' R
109eqimss2i 3149 . . 3  |-  ran  `' R  C_  dom  R
11 cores 5037 . . 3  |-  ( ran  `' R  C_  dom  R  ->  ( ( S  |`  dom  R )  o.  `' R )  =  ( S  o.  `' R
) )
1210, 11ax-mp 5 . 2  |-  ( ( S  |`  dom  R )  o.  `' R )  =  ( S  o.  `' R )
135, 8, 123eqtr3ri 2167 1  |-  ( S  o.  `' R )  =  ( ( S  |`  dom  R )  o.  `' ( R  |`  dom  S ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1331    C_ wss 3066   `'ccnv 4533   dom cdm 4534   ran crn 4535    |` cres 4536    o. ccom 4538
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546
This theorem is referenced by:  cocnvss  5059
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