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Metamath Proof Explorer

Theorem 1cossres

Description: The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019)

Ref Expression
Assertion 1cossres 
|- ,~ ( R |` A ) = { <. x , y >. | E. u e. A ( u R x /\ u R y ) }

Proof

Step Hyp Ref Expression
1 df-coss 
 |-  ,~ ( R |` A ) = { <. x , y >. | E. u ( u ( R |` A ) x /\ u ( R |` A ) y ) }
2 df-rex 
 |-  ( E. u e. A ( u R x /\ u R y ) <-> E. u ( u e. A /\ ( u R x /\ u R y ) ) )
3 anandi 
 |-  ( ( u e. A /\ ( u R x /\ u R y ) ) <-> ( ( u e. A /\ u R x ) /\ ( u e. A /\ u R y ) ) )
4 brres 
 |-  ( x e. _V -> ( u ( R |` A ) x <-> ( u e. A /\ u R x ) ) )
5 4 elv 
 |-  ( u ( R |` A ) x <-> ( u e. A /\ u R x ) )
6 brres 
 |-  ( y e. _V -> ( u ( R |` A ) y <-> ( u e. A /\ u R y ) ) )
7 6 elv 
 |-  ( u ( R |` A ) y <-> ( u e. A /\ u R y ) )
8 5 7 anbi12i 
 |-  ( ( u ( R |` A ) x /\ u ( R |` A ) y ) <-> ( ( u e. A /\ u R x ) /\ ( u e. A /\ u R y ) ) )
9 3 8 bitr4i 
 |-  ( ( u e. A /\ ( u R x /\ u R y ) ) <-> ( u ( R |` A ) x /\ u ( R |` A ) y ) )
10 9 exbii 
 |-  ( E. u ( u e. A /\ ( u R x /\ u R y ) ) <-> E. u ( u ( R |` A ) x /\ u ( R |` A ) y ) )
11 2 10 bitri 
 |-  ( E. u e. A ( u R x /\ u R y ) <-> E. u ( u ( R |` A ) x /\ u ( R |` A ) y ) )
12 11 opabbii 
 |-  { <. x , y >. | E. u e. A ( u R x /\ u R y ) } = { <. x , y >. | E. u ( u ( R |` A ) x /\ u ( R |` A ) y ) }
13 1 12 eqtr4i 
 |-  ,~ ( R |` A ) = { <. x , y >. | E. u e. A ( u R x /\ u R y ) }