Metamath Proof Explorer


Theorem dfrefrel2

Description: Alternate definition of the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021)

Ref Expression
Assertion dfrefrel2
|- ( RefRel R <-> ( ( _I i^i ( dom R X. ran R ) ) C_ R /\ Rel R ) )

Proof

Step Hyp Ref Expression
1 df-refrel
 |-  ( RefRel R <-> ( ( _I 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 sseq2d
 |-  ( Rel R -> ( ( _I i^i ( dom R X. ran R ) ) C_ ( R i^i ( dom R X. ran R ) ) <-> ( _I i^i ( dom R X. ran R ) ) C_ R ) )
5 4 pm5.32ri
 |-  ( ( ( _I i^i ( dom R X. ran R ) ) C_ ( R i^i ( dom R X. ran R ) ) /\ Rel R ) <-> ( ( _I i^i ( dom R X. ran R ) ) C_ R /\ Rel R ) )
6 1 5 bitri
 |-  ( RefRel R <-> ( ( _I i^i ( dom R X. ran R ) ) C_ R /\ Rel R ) )