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Metamath Proof Explorer

Theorem trressn

Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 ) is transitive, see also trrelressn . (Contributed by Peter Mazsa, 16-Jun-2024)

Ref Expression
Assertion trressn 
|- A. x A. y A. z ( ( x ( R |` { A } ) y /\ y ( R |` { A } ) z ) -> x ( R |` { A } ) z )

Proof

Step Hyp Ref Expression
1 an3 
 |-  ( ( ( x = A /\ A R y ) /\ ( y = A /\ A R z ) ) -> ( x = A /\ A R z ) )
2 eqbrb 
 |-  ( ( x = A /\ x R y ) <-> ( x = A /\ A R y ) )
3 eqbrb 
 |-  ( ( y = A /\ y R z ) <-> ( y = A /\ A R z ) )
4 2 3 anbi12i 
 |-  ( ( ( x = A /\ x R y ) /\ ( y = A /\ y R z ) ) <-> ( ( x = A /\ A R y ) /\ ( y = A /\ A R z ) ) )
5 eqbrb 
 |-  ( ( x = A /\ x R z ) <-> ( x = A /\ A R z ) )
6 1 4 5 3imtr4i 
 |-  ( ( ( x = A /\ x R y ) /\ ( y = A /\ y R z ) ) -> ( x = A /\ x R z ) )
7 brressn 
 |-  ( ( x e. _V /\ y e. _V ) -> ( x ( R |` { A } ) y <-> ( x = A /\ x R y ) ) )
8 7 el2v 
 |-  ( x ( R |` { A } ) y <-> ( x = A /\ x R y ) )
9 brressn 
 |-  ( ( y e. _V /\ z e. _V ) -> ( y ( R |` { A } ) z <-> ( y = A /\ y R z ) ) )
10 9 el2v 
 |-  ( y ( R |` { A } ) z <-> ( y = A /\ y R z ) )
11 8 10 anbi12i 
 |-  ( ( x ( R |` { A } ) y /\ y ( R |` { A } ) z ) <-> ( ( x = A /\ x R y ) /\ ( y = A /\ y R z ) ) )
12 brressn 
 |-  ( ( x e. _V /\ z e. _V ) -> ( x ( R |` { A } ) z <-> ( x = A /\ x R z ) ) )
13 12 el2v 
 |-  ( x ( R |` { A } ) z <-> ( x = A /\ x R z ) )
14 6 11 13 3imtr4i 
 |-  ( ( x ( R |` { A } ) y /\ y ( R |` { A } ) z ) -> x ( R |` { A } ) z )
15 14 gen2 
 |-  A. y A. z ( ( x ( R |` { A } ) y /\ y ( R |` { A } ) z ) -> x ( R |` { A } ) z )
16 15 ax-gen 
 |-  A. x A. y A. z ( ( x ( R |` { A } ) y /\ y ( R |` { A } ) z ) -> x ( R |` { A } ) z )