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Theorem cocnvres 4955
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 4737 . . . 4  |-  ( S  |`  dom  R )  C_  S
2 dmss 4635 . . . 4  |-  ( ( S  |`  dom  R ) 
C_  S  ->  dom  ( S  |`  dom  R
)  C_  dom  S )
31, 2ax-mp 7 . . 3  |-  dom  ( S  |`  dom  R ) 
C_  dom  S
4 cores2 4943 . . 3  |-  ( dom  ( S  |`  dom  R
)  C_  dom  S  -> 
( ( S  |`  dom  R )  o.  `' ( `' `' R  |`  dom  S
) )  =  ( ( S  |`  dom  R
)  o.  `' R
) )
53, 4ax-mp 7 . 2  |-  ( ( S  |`  dom  R )  o.  `' ( `' `' R  |`  dom  S
) )  =  ( ( S  |`  dom  R
)  o.  `' R
)
6 rescnvcnv 4893 . . . 4  |-  ( `' `' R  |`  dom  S
)  =  ( R  |`  dom  S )
76cnveqi 4611 . . 3  |-  `' ( `' `' R  |`  dom  S
)  =  `' ( R  |`  dom  S )
87coeq2i 4596 . 2  |-  ( ( S  |`  dom  R )  o.  `' ( `' `' R  |`  dom  S
) )  =  ( ( S  |`  dom  R
)  o.  `' ( R  |`  dom  S ) )
9 dfdm4 4628 . . . 4  |-  dom  R  =  ran  `' R
109eqimss2i 3081 . . 3  |-  ran  `' R  C_  dom  R
11 cores 4934 . . 3  |-  ( ran  `' R  C_  dom  R  ->  ( ( S  |`  dom  R )  o.  `' R )  =  ( S  o.  `' R
) )
1210, 11ax-mp 7 . 2  |-  ( ( S  |`  dom  R )  o.  `' R )  =  ( S  o.  `' R )
135, 8, 123eqtr3ri 2117 1  |-  ( S  o.  `' R )  =  ( ( S  |`  dom  R )  o.  `' ( R  |`  dom  S ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1289    C_ wss 2999   `'ccnv 4437   dom cdm 4438   ran crn 4439    |` cres 4440    o. ccom 4442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450
This theorem is referenced by:  cocnvss  4956
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