ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cocnvres Unicode version

Theorem cocnvres 5171
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 4949 . . . 4  |-  ( S  |`  dom  R )  C_  S
2 dmss 4844 . . . 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 5159 . . 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 5109 . . . 4  |-  ( `' `' R  |`  dom  S
)  =  ( R  |`  dom  S )
76cnveqi 4820 . . 3  |-  `' ( `' `' R  |`  dom  S
)  =  `' ( R  |`  dom  S )
87coeq2i 4805 . 2  |-  ( ( S  |`  dom  R )  o.  `' ( `' `' R  |`  dom  S
) )  =  ( ( S  |`  dom  R
)  o.  `' ( R  |`  dom  S ) )
9 dfdm4 4837 . . . 4  |-  dom  R  =  ran  `' R
109eqimss2i 3227 . . 3  |-  ran  `' R  C_  dom  R
11 cores 5150 . . 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 2219 1  |-  ( S  o.  `' R )  =  ( ( S  |`  dom  R )  o.  `' ( R  |`  dom  S ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1364    C_ wss 3144   `'ccnv 4643   dom cdm 4644   ran crn 4645    |` cres 4646    o. ccom 4648
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656
This theorem is referenced by:  cocnvss  5172
  Copyright terms: Public domain W3C validator