Metamath Proof Explorer


Definition df-cnvrefrel

Description: Define the converse reflexive relation predicate (read: R is a converse reflexive relation), see also the comment of dfcnvrefrel3 . Alternate definitions are dfcnvrefrel2 and dfcnvrefrel3 . (Contributed by Peter Mazsa, 16-Jul-2021)

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

Detailed syntax breakdown

Step Hyp Ref Expression
0 cR
 |-  R
1 0 wcnvrefrel
 |-  CnvRefRel R
2 0 cdm
 |-  dom R
3 0 crn
 |-  ran R
4 2 3 cxp
 |-  ( dom R X. ran R )
5 0 4 cin
 |-  ( R i^i ( dom R X. ran R ) )
6 cid
 |-  _I
7 6 4 cin
 |-  ( _I i^i ( dom R X. ran R ) )
8 5 7 wss
 |-  ( R i^i ( dom R X. ran R ) ) C_ ( _I i^i ( dom R X. ran R ) )
9 0 wrel
 |-  Rel R
10 8 9 wa
 |-  ( ( R i^i ( dom R X. ran R ) ) C_ ( _I i^i ( dom R X. ran R ) ) /\ Rel R )
11 1 10 wb
 |-  ( CnvRefRel R <-> ( ( R i^i ( dom R X. ran R ) ) C_ ( _I i^i ( dom R X. ran R ) ) /\ Rel R ) )