Metamath Proof Explorer

Theorem disjres

Description: Disjoint restriction. (Contributed by Peter Mazsa, 25-Aug-2023)

Ref Expression
Assertion disjres 
|- ( Rel R -> ( Disj ( R |` A ) <-> A. u e. A A. v e. A ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) ) )

Proof

Step Hyp Ref Expression
1 relres 
 |-  Rel ( R |` A )
2 dfdisjALTV4 
 |-  ( Disj ( R |` A ) <-> ( A. x E* u u ( R |` A ) x /\ Rel ( R |` A ) ) )
3 1 2 mpbiran2 
 |-  ( Disj ( R |` A ) <-> A. x E* u u ( R |` A ) x )
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 5 mobii 
 |-  ( E* u u ( R |` A ) x <-> E* u ( u e. A /\ u R x ) )
7 df-rmo 
 |-  ( E* u e. A u R x <-> E* u ( u e. A /\ u R x ) )
8 6 7 bitr4i 
 |-  ( E* u u ( R |` A ) x <-> E* u e. A u R x )
9 8 albii 
 |-  ( A. x E* u u ( R |` A ) x <-> A. x E* u e. A u R x )
10 3 9 bitri 
 |-  ( Disj ( R |` A ) <-> A. x E* u e. A u R x )
11 id 
 |-  ( u = v -> u = v )
12 11 inecmo 
 |-  ( Rel R -> ( A. u e. A A. v e. A ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) <-> A. x E* u e. A u R x ) )
13 10 12 bitr4id 
 |-  ( Rel R -> ( Disj ( R |` A ) <-> A. u e. A A. v e. A ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) ) )