Metamath Proof Explorer


Theorem upgr1wlkdlem1

Description: Lemma 1 for upgr1wlkd . (Contributed by AV, 22-Jan-2021)

Ref Expression
Hypotheses upgr1wlkd.p
|- P = <" X Y ">
upgr1wlkd.f
|- F = <" J ">
upgr1wlkd.x
|- ( ph -> X e. ( Vtx ` G ) )
upgr1wlkd.y
|- ( ph -> Y e. ( Vtx ` G ) )
upgr1wlkd.j
|- ( ph -> ( ( iEdg ` G ) ` J ) = { X , Y } )
Assertion upgr1wlkdlem1
|- ( ( ph /\ X = Y ) -> ( ( iEdg ` G ) ` J ) = { X } )

Proof

Step Hyp Ref Expression
1 upgr1wlkd.p
 |-  P = <" X Y ">
2 upgr1wlkd.f
 |-  F = <" J ">
3 upgr1wlkd.x
 |-  ( ph -> X e. ( Vtx ` G ) )
4 upgr1wlkd.y
 |-  ( ph -> Y e. ( Vtx ` G ) )
5 upgr1wlkd.j
 |-  ( ph -> ( ( iEdg ` G ) ` J ) = { X , Y } )
6 preq2
 |-  ( Y = X -> { X , Y } = { X , X } )
7 6 eqeq2d
 |-  ( Y = X -> ( ( ( iEdg ` G ) ` J ) = { X , Y } <-> ( ( iEdg ` G ) ` J ) = { X , X } ) )
8 7 eqcoms
 |-  ( X = Y -> ( ( ( iEdg ` G ) ` J ) = { X , Y } <-> ( ( iEdg ` G ) ` J ) = { X , X } ) )
9 simpl
 |-  ( ( ( ( iEdg ` G ) ` J ) = { X , X } /\ ph ) -> ( ( iEdg ` G ) ` J ) = { X , X } )
10 dfsn2
 |-  { X } = { X , X }
11 9 10 eqtr4di
 |-  ( ( ( ( iEdg ` G ) ` J ) = { X , X } /\ ph ) -> ( ( iEdg ` G ) ` J ) = { X } )
12 11 ex
 |-  ( ( ( iEdg ` G ) ` J ) = { X , X } -> ( ph -> ( ( iEdg ` G ) ` J ) = { X } ) )
13 8 12 syl6bi
 |-  ( X = Y -> ( ( ( iEdg ` G ) ` J ) = { X , Y } -> ( ph -> ( ( iEdg ` G ) ` J ) = { X } ) ) )
14 13 com13
 |-  ( ph -> ( ( ( iEdg ` G ) ` J ) = { X , Y } -> ( X = Y -> ( ( iEdg ` G ) ` J ) = { X } ) ) )
15 5 14 mpd
 |-  ( ph -> ( X = Y -> ( ( iEdg ` G ) ` J ) = { X } ) )
16 15 imp
 |-  ( ( ph /\ X = Y ) -> ( ( iEdg ` G ) ` J ) = { X } )