Step |
Hyp |
Ref |
Expression |
1 |
|
umgr2cycllem.1 |
|- F = <" J K "> |
2 |
|
umgr2cycllem.2 |
|- I = ( iEdg ` G ) |
3 |
|
umgr2cycllem.3 |
|- ( ph -> G e. UMGraph ) |
4 |
|
umgr2cycllem.4 |
|- ( ph -> J e. dom I ) |
5 |
|
umgr2cycllem.5 |
|- ( ph -> J =/= K ) |
6 |
|
umgr2cycllem.6 |
|- ( ph -> ( I ` J ) = ( I ` K ) ) |
7 |
|
umgruhgr |
|- ( G e. UMGraph -> G e. UHGraph ) |
8 |
2
|
uhgrfun |
|- ( G e. UHGraph -> Fun I ) |
9 |
3 7 8
|
3syl |
|- ( ph -> Fun I ) |
10 |
2
|
iedgedg |
|- ( ( Fun I /\ J e. dom I ) -> ( I ` J ) e. ( Edg ` G ) ) |
11 |
9 4 10
|
syl2anc |
|- ( ph -> ( I ` J ) e. ( Edg ` G ) ) |
12 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
13 |
|
eqid |
|- ( Edg ` G ) = ( Edg ` G ) |
14 |
12 13
|
umgredg |
|- ( ( G e. UMGraph /\ ( I ` J ) e. ( Edg ` G ) ) -> E. a e. ( Vtx ` G ) E. b e. ( Vtx ` G ) ( a =/= b /\ ( I ` J ) = { a , b } ) ) |
15 |
3 11 14
|
syl2anc |
|- ( ph -> E. a e. ( Vtx ` G ) E. b e. ( Vtx ` G ) ( a =/= b /\ ( I ` J ) = { a , b } ) ) |
16 |
|
ax-5 |
|- ( a e. ( Vtx ` G ) -> A. b a e. ( Vtx ` G ) ) |
17 |
|
alral |
|- ( A. b a e. ( Vtx ` G ) -> A. b e. ( Vtx ` G ) a e. ( Vtx ` G ) ) |
18 |
16 17
|
syl |
|- ( a e. ( Vtx ` G ) -> A. b e. ( Vtx ` G ) a e. ( Vtx ` G ) ) |
19 |
|
r19.29 |
|- ( ( A. b e. ( Vtx ` G ) a e. ( Vtx ` G ) /\ E. b e. ( Vtx ` G ) ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> E. b e. ( Vtx ` G ) ( a e. ( Vtx ` G ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) ) |
20 |
18 19
|
sylan |
|- ( ( a e. ( Vtx ` G ) /\ E. b e. ( Vtx ` G ) ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> E. b e. ( Vtx ` G ) ( a e. ( Vtx ` G ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) ) |
21 |
|
eqid |
|- <" a b a "> = <" a b a "> |
22 |
|
simp2 |
|- ( ( ph /\ ( a e. ( Vtx ` G ) /\ b e. ( Vtx ` G ) ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> ( a e. ( Vtx ` G ) /\ b e. ( Vtx ` G ) ) ) |
23 |
|
simp3l |
|- ( ( ph /\ ( a e. ( Vtx ` G ) /\ b e. ( Vtx ` G ) ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> a =/= b ) |
24 |
|
eqimss2 |
|- ( ( I ` J ) = { a , b } -> { a , b } C_ ( I ` J ) ) |
25 |
24
|
adantl |
|- ( ( a =/= b /\ ( I ` J ) = { a , b } ) -> { a , b } C_ ( I ` J ) ) |
26 |
25
|
3ad2ant3 |
|- ( ( ph /\ ( a e. ( Vtx ` G ) /\ b e. ( Vtx ` G ) ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> { a , b } C_ ( I ` J ) ) |
27 |
6
|
sseq2d |
|- ( ph -> ( { a , b } C_ ( I ` J ) <-> { a , b } C_ ( I ` K ) ) ) |
28 |
24 27
|
syl5ib |
|- ( ph -> ( ( I ` J ) = { a , b } -> { a , b } C_ ( I ` K ) ) ) |
29 |
28
|
adantld |
|- ( ph -> ( ( a =/= b /\ ( I ` J ) = { a , b } ) -> { a , b } C_ ( I ` K ) ) ) |
30 |
29
|
adantld |
|- ( ph -> ( ( ( a e. ( Vtx ` G ) /\ b e. ( Vtx ` G ) ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> { a , b } C_ ( I ` K ) ) ) |
31 |
30
|
3impib |
|- ( ( ph /\ ( a e. ( Vtx ` G ) /\ b e. ( Vtx ` G ) ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> { a , b } C_ ( I ` K ) ) |
32 |
26 31
|
jca |
|- ( ( ph /\ ( a e. ( Vtx ` G ) /\ b e. ( Vtx ` G ) ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> ( { a , b } C_ ( I ` J ) /\ { a , b } C_ ( I ` K ) ) ) |
33 |
5
|
3ad2ant1 |
|- ( ( ph /\ ( a e. ( Vtx ` G ) /\ b e. ( Vtx ` G ) ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> J =/= K ) |
34 |
21 1 22 23 32 12 2 33
|
2cycl2d |
|- ( ( ph /\ ( a e. ( Vtx ` G ) /\ b e. ( Vtx ` G ) ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> F ( Cycles ` G ) <" a b a "> ) |
35 |
34
|
3expib |
|- ( ph -> ( ( ( a e. ( Vtx ` G ) /\ b e. ( Vtx ` G ) ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> F ( Cycles ` G ) <" a b a "> ) ) |
36 |
35
|
exp4c |
|- ( ph -> ( a e. ( Vtx ` G ) -> ( b e. ( Vtx ` G ) -> ( ( a =/= b /\ ( I ` J ) = { a , b } ) -> F ( Cycles ` G ) <" a b a "> ) ) ) ) |
37 |
36
|
com23 |
|- ( ph -> ( b e. ( Vtx ` G ) -> ( a e. ( Vtx ` G ) -> ( ( a =/= b /\ ( I ` J ) = { a , b } ) -> F ( Cycles ` G ) <" a b a "> ) ) ) ) |
38 |
37
|
imp4a |
|- ( ph -> ( b e. ( Vtx ` G ) -> ( ( a e. ( Vtx ` G ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> F ( Cycles ` G ) <" a b a "> ) ) ) |
39 |
|
s3cli |
|- <" a b a "> e. Word _V |
40 |
|
breq2 |
|- ( p = <" a b a "> -> ( F ( Cycles ` G ) p <-> F ( Cycles ` G ) <" a b a "> ) ) |
41 |
40
|
rspcev |
|- ( ( <" a b a "> e. Word _V /\ F ( Cycles ` G ) <" a b a "> ) -> E. p e. Word _V F ( Cycles ` G ) p ) |
42 |
39 41
|
mpan |
|- ( F ( Cycles ` G ) <" a b a "> -> E. p e. Word _V F ( Cycles ` G ) p ) |
43 |
|
rexex |
|- ( E. p e. Word _V F ( Cycles ` G ) p -> E. p F ( Cycles ` G ) p ) |
44 |
42 43
|
syl |
|- ( F ( Cycles ` G ) <" a b a "> -> E. p F ( Cycles ` G ) p ) |
45 |
38 44
|
syl8 |
|- ( ph -> ( b e. ( Vtx ` G ) -> ( ( a e. ( Vtx ` G ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> E. p F ( Cycles ` G ) p ) ) ) |
46 |
45
|
rexlimdv |
|- ( ph -> ( E. b e. ( Vtx ` G ) ( a e. ( Vtx ` G ) /\ ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> E. p F ( Cycles ` G ) p ) ) |
47 |
20 46
|
syl5 |
|- ( ph -> ( ( a e. ( Vtx ` G ) /\ E. b e. ( Vtx ` G ) ( a =/= b /\ ( I ` J ) = { a , b } ) ) -> E. p F ( Cycles ` G ) p ) ) |
48 |
47
|
expd |
|- ( ph -> ( a e. ( Vtx ` G ) -> ( E. b e. ( Vtx ` G ) ( a =/= b /\ ( I ` J ) = { a , b } ) -> E. p F ( Cycles ` G ) p ) ) ) |
49 |
48
|
rexlimdv |
|- ( ph -> ( E. a e. ( Vtx ` G ) E. b e. ( Vtx ` G ) ( a =/= b /\ ( I ` J ) = { a , b } ) -> E. p F ( Cycles ` G ) p ) ) |
50 |
15 49
|
mpd |
|- ( ph -> E. p F ( Cycles ` G ) p ) |