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Metamath Proof Explorer

Theorem prtlem19

Description: Lemma for prter2 . (Contributed by Rodolfo Medina, 15-Oct-2010) (Revised by Mario Carneiro, 12-Aug-2015)

Ref Expression
Hypothesis prtlem18.1 
|- .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }
Assertion prtlem19 
|- ( Prt A -> ( ( v e. A /\ z e. v ) -> v = [ z ] .~ ) )

Proof

Step Hyp Ref Expression
1 prtlem18.1 
 |-  .~ = { <. x , y >. | E. u e. A ( x e. u /\ y e. u ) }
2 1 prtlem18 
 |-  ( Prt A -> ( ( v e. A /\ z e. v ) -> ( w e. v <-> z .~ w ) ) )
3 2 imp 
 |-  ( ( Prt A /\ ( v e. A /\ z e. v ) ) -> ( w e. v <-> z .~ w ) )
4 vex 
 |-  w e. _V
5 vex 
 |-  z e. _V
6 4 5 elec 
 |-  ( w e. [ z ] .~ <-> z .~ w )
7 3 6 bitr4di 
 |-  ( ( Prt A /\ ( v e. A /\ z e. v ) ) -> ( w e. v <-> w e. [ z ] .~ ) )
8 7 eqrdv 
 |-  ( ( Prt A /\ ( v e. A /\ z e. v ) ) -> v = [ z ] .~ )
9 8 ex 
 |-  ( Prt A -> ( ( v e. A /\ z e. v ) -> v = [ z ] .~ ) )