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Metamath Proof Explorer

Theorem br1cnvres

Description: Binary relation on the converse of a restriction. (Contributed by Peter Mazsa, 27-Jul-2019)

Ref Expression
Assertion br1cnvres 
|- ( B e. V -> ( B `' ( R |` A ) C <-> ( C e. A /\ C R B ) ) )

Proof

Step Hyp Ref Expression
1 df-res 
 |-  ( R |` A ) = ( R i^i ( A X. _V ) )
2 1 cnveqi 
 |-  `' ( R |` A ) = `' ( R i^i ( A X. _V ) )
3 2 breqi 
 |-  ( B `' ( R |` A ) C <-> B `' ( R i^i ( A X. _V ) ) C )
4 elex 
 |-  ( B e. V -> B e. _V )
5 br1cnvinxp 
 |-  ( B `' ( R i^i ( A X. _V ) ) C <-> ( ( B e. _V /\ C e. A ) /\ C R B ) )
6 anass 
 |-  ( ( ( B e. _V /\ C e. A ) /\ C R B ) <-> ( B e. _V /\ ( C e. A /\ C R B ) ) )
7 5 6 bitri 
 |-  ( B `' ( R i^i ( A X. _V ) ) C <-> ( B e. _V /\ ( C e. A /\ C R B ) ) )
8 7 baib 
 |-  ( B e. _V -> ( B `' ( R i^i ( A X. _V ) ) C <-> ( C e. A /\ C R B ) ) )
9 4 8 syl 
 |-  ( B e. V -> ( B `' ( R i^i ( A X. _V ) ) C <-> ( C e. A /\ C R B ) ) )
10 3 9 bitrid 
 |-  ( B e. V -> ( B `' ( R |` A ) C <-> ( C e. A /\ C R B ) ) )