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Metamath Proof Explorer

Theorem dftrrel2

Description: Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021)

Ref Expression
Assertion dftrrel2 
|- ( TrRel R <-> ( ( R o. R ) C_ R /\ Rel R ) )

Proof

Step Hyp Ref Expression
1 df-trrel 
 |-  ( TrRel R <-> ( ( ( R i^i ( dom R X. ran R ) ) o. ( R i^i ( dom R X. ran R ) ) ) C_ ( R i^i ( dom R X. ran R ) ) /\ Rel R ) )
2 dfrel6 
 |-  ( Rel R <-> ( R i^i ( dom R X. ran R ) ) = R )
3 2 biimpi 
 |-  ( Rel R -> ( R i^i ( dom R X. ran R ) ) = R )
4 3 3 coeq12d 
 |-  ( Rel R -> ( ( R i^i ( dom R X. ran R ) ) o. ( R i^i ( dom R X. ran R ) ) ) = ( R o. R ) )
5 4 3 sseq12d 
 |-  ( Rel R -> ( ( ( R i^i ( dom R X. ran R ) ) o. ( R i^i ( dom R X. ran R ) ) ) C_ ( R i^i ( dom R X. ran R ) ) <-> ( R o. R ) C_ R ) )
6 5 pm5.32ri 
 |-  ( ( ( ( R i^i ( dom R X. ran R ) ) o. ( R i^i ( dom R X. ran R ) ) ) C_ ( R i^i ( dom R X. ran R ) ) /\ Rel R ) <-> ( ( R o. R ) C_ R /\ Rel R ) )
7 1 6 bitri 
 |-  ( TrRel R <-> ( ( R o. R ) C_ R /\ Rel R ) )