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Metamath Proof Explorer

Theorem dfcnvrefrel2

Description: Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 24-Jul-2019)

Ref Expression
Assertion dfcnvrefrel2 
|- ( CnvRefRel R <-> ( R C_ ( _I i^i ( dom R X. ran R ) ) /\ Rel R ) )

Proof

Step Hyp Ref Expression
1 df-cnvrefrel 
 |-  ( CnvRefRel R <-> ( ( R i^i ( dom R X. ran R ) ) C_ ( _I 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 sseq1d 
 |-  ( Rel R -> ( ( R i^i ( dom R X. ran R ) ) C_ ( _I i^i ( dom R X. ran R ) ) <-> R C_ ( _I i^i ( dom R X. ran R ) ) ) )
5 4 pm5.32ri 
 |-  ( ( ( R i^i ( dom R X. ran R ) ) C_ ( _I i^i ( dom R X. ran R ) ) /\ Rel R ) <-> ( R C_ ( _I i^i ( dom R X. ran R ) ) /\ Rel R ) )
6 1 5 bitri 
 |-  ( CnvRefRel R <-> ( R C_ ( _I i^i ( dom R X. ran R ) ) /\ Rel R ) )