| 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
|
biimtrdi |
|- ( 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 } ) |