Metamath Proof Explorer

Theorem coss1cnvres

Description: Class of cosets by the converse of a restriction. (Contributed by Peter Mazsa, 8-Jun-2020)

Ref Expression
Assertion coss1cnvres 
|- ,~ `' ( R |` A ) = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ E. x ( u R x /\ v R x ) ) }

Proof

Step Hyp Ref Expression
1 df-coss 
 |-  ,~ `' ( R |` A ) = { <. u , v >. | E. x ( x `' ( R |` A ) u /\ x `' ( R |` A ) v ) }
2 br1cnvres 
 |-  ( x e. _V -> ( x `' ( R |` A ) u <-> ( u e. A /\ u R x ) ) )
3 2 elv 
 |-  ( x `' ( R |` A ) u <-> ( u e. A /\ u R x ) )
4 br1cnvres 
 |-  ( x e. _V -> ( x `' ( R |` A ) v <-> ( v e. A /\ v R x ) ) )
5 4 elv 
 |-  ( x `' ( R |` A ) v <-> ( v e. A /\ v R x ) )
6 3 5 anbi12i 
 |-  ( ( x `' ( R |` A ) u /\ x `' ( R |` A ) v ) <-> ( ( u e. A /\ u R x ) /\ ( v e. A /\ v R x ) ) )
7 an4 
 |-  ( ( ( u e. A /\ v e. A ) /\ ( u R x /\ v R x ) ) <-> ( ( u e. A /\ u R x ) /\ ( v e. A /\ v R x ) ) )
8 6 7 bitr4i 
 |-  ( ( x `' ( R |` A ) u /\ x `' ( R |` A ) v ) <-> ( ( u e. A /\ v e. A ) /\ ( u R x /\ v R x ) ) )
9 8 exbii 
 |-  ( E. x ( x `' ( R |` A ) u /\ x `' ( R |` A ) v ) <-> E. x ( ( u e. A /\ v e. A ) /\ ( u R x /\ v R x ) ) )
10 19.42v 
 |-  ( E. x ( ( u e. A /\ v e. A ) /\ ( u R x /\ v R x ) ) <-> ( ( u e. A /\ v e. A ) /\ E. x ( u R x /\ v R x ) ) )
11 9 10 bitri 
 |-  ( E. x ( x `' ( R |` A ) u /\ x `' ( R |` A ) v ) <-> ( ( u e. A /\ v e. A ) /\ E. x ( u R x /\ v R x ) ) )
12 11 opabbii 
 |-  { <. u , v >. | E. x ( x `' ( R |` A ) u /\ x `' ( R |` A ) v ) } = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ E. x ( u R x /\ v R x ) ) }
13 1 12 eqtri 
 |-  ,~ `' ( R |` A ) = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ E. x ( u R x /\ v R x ) ) }