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Theorem upgrwlkupwlk 41721
Description: In a pseudograph, a walk is a simple walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.)
Assertion
Ref Expression
upgrwlkupwlk |- ( ( G e. UPGraph /\ F (Walks ` G ) P ) -> F (UPWalks ` G ) P )
Proof of Theorem upgrwlkupwlk
Dummy variable k is distinct from all other variables.
StepHypRef Expression
1 wlkv 26508 . . 3 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
2 eqid 2622 . . . . . . . . 9 |- (Vtx ` G ) = (Vtx ` G )
3 eqid 2622 . . . . . . . . 9 |- (iEdg ` G ) = (iEdg ` G )
42, 3iswlk 26506 . . . . . . . 8 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F (Walks ` G ) P <-> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) ) )
5 simpr1 1067 . . . . . . . . . . 11 |- ( ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) ) -> F e. Word dom (iEdg ` G ) )
6 simpr2 1068 . . . . . . . . . . 11 |- ( ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) ) -> P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) )
7 df-ifp 1013 . . . . . . . . . . . . . . . . 17 |- (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } ) \/ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) )
8 dfsn2 4190 . . . . . . . . . . . . . . . . . . . . . . 23 |- { ( P ` k ) } = { ( P ` k ) , ( P ` k ) }
9 preq2 4269 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> { ( P ` k ) , ( P ` k ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
108, 9syl5eq 2668 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> { ( P ` k ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
1110eqeq2d 2632 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> ( ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } <-> ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
1211biimpa 501 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } ) -> ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
1312a1d 25 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } ) -> ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
14 simpr 477 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) -> G e. UPGraph )
15 simpl 473 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> F e. Word dom (iEdg ` G ) )
16 eqid 2622 . . . . . . . . . . . . . . . . . . . . . . . 24 |- (Edg ` G ) = (Edg ` G )
173, 16upgredginwlk 26532 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( G e. UPGraph /\ F e. Word dom (iEdg ` G ) ) -> ( k e. ( 0..^ ( # ` F ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) ) )
1814, 15, 17syl2anr 495 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) -> ( k e. ( 0..^ ( # ` F ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) ) )
1918imp 445 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) )
20 simprr 796 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) -> G e. UPGraph )
2120adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) -> G e. UPGraph )
2221adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) ) -> G e. UPGraph )
2322adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> G e. UPGraph )
24 simplr 792 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) )
25 simprr 796 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) )
26 df-ne 2795 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> -. ( P ` k ) = ( P ` ( k + 1 ) ) )
27 fvexd 6203 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( P ` k ) e. _V )
28 fvexd 6203 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( P ` ( k + 1 ) ) e. _V )
29 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) )
3027, 28, 293jca 1242 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( ( P ` k ) e. _V /\ ( P ` ( k + 1 ) ) e. _V /\ ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
3126, 30sylbir 225 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( -. ( P ` k ) = ( P ` ( k + 1 ) ) -> ( ( P ` k ) e. _V /\ ( P ` ( k + 1 ) ) e. _V /\ ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
3231adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) -> ( ( P ` k ) e. _V /\ ( P ` ( k + 1 ) ) e. _V /\ ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
3332adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> ( ( P ` k ) e. _V /\ ( P ` ( k + 1 ) ) e. _V /\ ( P ` k ) =/= ( P ` ( k + 1 ) ) ) )
342, 16upgredgpr 26037 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( G e. UPGraph /\ ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) /\ ( ( P ` k ) e. _V /\ ( P ` ( k + 1 ) ) e. _V /\ ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = ( (iEdg ` G ) ` ( F ` k ) ) )
3523, 24, 25, 33, 34syl31anc 1329 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = ( (iEdg ` G ) ` ( F ` k ) ) )
3635eqcomd 2628 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
3736exp31 630 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) -> ( ( (iEdg ` G ) ` ( F ` k ) ) e. (Edg ` G ) -> ( ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
3819, 37mpd 15 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) -> ( ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
3938com12 32 . . . . . . . . . . . . . . . . . . 19 |- ( ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) -> ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
4013, 39jaoi 394 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } ) \/ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
4140com12 32 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) -> ( ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } ) \/ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
427, 41syl5bi 232 . . . . . . . . . . . . . . . 16 |- ( ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0..^ ( # ` F ) ) ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) -> ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
4342ralimdva 2962 . . . . . . . . . . . . . . 15 |- ( ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) -> A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
4443ex 450 . . . . . . . . . . . . . 14 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) -> A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
4544com23 86 . . . . . . . . . . . . 13 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) -> ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) -> A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
46453impia 1261 . . . . . . . . . . . 12 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) -> A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
4746impcom 446 . . . . . . . . . . 11 |- ( ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) ) -> A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
485, 6, 473jca 1242 . . . . . . . . . 10 |- ( ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) ) -> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
4948exp31 630 . . . . . . . . 9 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( G e. UPGraph -> ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) )
5049com23 86 . . . . . . . 8 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> ( G e. UPGraph -> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) )
514, 50sylbid 230 . . . . . . 7 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F (Walks ` G ) P -> ( G e. UPGraph -> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) )
5251impd 447 . . . . . 6 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( ( F (Walks ` G ) P /\ G e. UPGraph ) -> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
5352impcom 446 . . . . 5 |- ( ( ( F (Walks ` G ) P /\ G e. UPGraph ) /\ ( G e. _V /\ F e. _V /\ P e. _V ) ) -> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
542, 3isupwlk 41717 . . . . . 6 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F (UPWalks ` G ) P <-> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
5554adantl 482 . . . . 5 |- ( ( ( F (Walks ` G ) P /\ G e. UPGraph ) /\ ( G e. _V /\ F e. _V /\ P e. _V ) ) -> ( F (UPWalks ` G ) P <-> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ ( # ` F ) ) ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
5653, 55mpbird 247 . . . 4 |- ( ( ( F (Walks ` G ) P /\ G e. UPGraph ) /\ ( G e. _V /\ F e. _V /\ P e. _V ) ) -> F (UPWalks ` G ) P )
5756exp31 630 . . 3 |- ( F (Walks ` G ) P -> ( G e. UPGraph -> ( ( G e. _V /\ F e. _V /\ P e. _V ) -> F (UPWalks ` G ) P ) ) )
581, 57mpid 44 . 2 |- ( F (Walks ` G ) P -> ( G e. UPGraph -> F (UPWalks ` G ) P ) )
5958impcom 446 1 |- ( ( G e. UPGraph /\ F (Walks ` G ) P ) -> F (UPWalks ` G ) P )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384  if-wif 1012   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   C_ wss 3574   {csn 4177   {cpr 4179   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UPGraph cupgr 25975  Walkscwlks 26492  UPWalkscupwlks 41714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-wlks 26495  df-upwlks 41715
This theorem is referenced by:  upgrwlkupwlkb  41722
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