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 } ) |