Theorem wlkp1lem4 26573

Description: Lemma for wlkp1 26578. (Contributed by AV, 6-Mar-2021.)

Hypotheses

Ref	Expression
wlkp1.v	|- V = (Vtx ` G )
wlkp1.i	|- I = (iEdg ` G )
wlkp1.f	|- ( ph -> Fun I )
wlkp1.a	|- ( ph -> I e. Fin )
wlkp1.b	|- ( ph -> B e. _V )
wlkp1.c	|- ( ph -> C e. V )
wlkp1.d	|- ( ph -> -. B e. dom I )
wlkp1.w	|- ( ph -> F (Walks ` G ) P )
wlkp1.n	|- N = ( # ` F )
wlkp1.e	|- ( ph -> E e. (Edg ` G ) )
wlkp1.x	|- ( ph -> { ( P ` N ) , C } C_ E )
wlkp1.u	|- ( ph -> (iEdg ` S ) = ( I u. { <. B , E >. } ) )
wlkp1.h	|- H = ( F u. { <. N , B >. } )
wlkp1.q	|- Q = ( P u. { <. ( N + 1 ) , C >. } )
wlkp1.s	|- ( ph -> (Vtx ` S ) = V )

Assertion

Ref	Expression
wlkp1lem4	|- ( ph -> ( S e. _V /\ H e. _V /\ Q e. _V ) )

Proof of Theorem wlkp1lem4

Step	Hyp	Ref	Expression
1		wlkp1.w	. . 3 |- ( ph -> F (Walks ` G ) P )
2		eqid 2622	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
3	2	wlkf 26510	. . . 4 |- ( F (Walks ` G ) P -> F e. Word dom (iEdg ` G ) )
4		eqid 2622	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
5	4	wlkp 26512	. . . 4 |- ( F (Walks ` G ) P -> P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) )
6	3, 5	jca 554	. . 3 |- ( F (Walks ` G ) P -> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) )
7	1, 6	syl 17	. 2 |- ( ph -> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) )
8		wlkp1.c	. . . . . 6 |- ( ph -> C e. V )
9		wlkp1.s	. . . . . 6 |- ( ph -> (Vtx ` S ) = V )
10	8, 9	eleqtrrd 2704	. . . . 5 |- ( ph -> C e. (Vtx ` S ) )
11	10	elfvexd 6222	. . . 4 |- ( ph -> S e. _V )
12	11	adantr 481	. . 3 |- ( ( ph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) ) -> S e. _V )
13		wlkp1.h	. . . 4 |- H = ( F u. { <. N , B >. } )
14		simprl 794	. . . . 5 |- ( ( ph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) ) -> F e. Word dom (iEdg ` G ) )
15		snex 4908	. . . . 5 |- { <. N , B >. } e. _V
16		unexg 6959	. . . . 5 |- ( ( F e. Word dom (iEdg ` G ) /\ { <. N , B >. } e. _V ) -> ( F u. { <. N , B >. } ) e. _V )
17	14, 15, 16	sylancl 694	. . . 4 |- ( ( ph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) ) -> ( F u. { <. N , B >. } ) e. _V )
18	13, 17	syl5eqel 2705	. . 3 |- ( ( ph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) ) -> H e. _V )
19		wlkp1.q	. . . 4 |- Q = ( P u. { <. ( N + 1 ) , C >. } )
20		ovex 6678	. . . . . . 7 |- ( 0 ... ( # ` F ) ) e. _V
21		fvex 6201	. . . . . . 7 |- (Vtx ` G ) e. _V
22	20, 21	fpm 7890	. . . . . 6 |- ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> P e. ( (Vtx ` G ) ^pm ( 0 ... ( # ` F ) ) ) )
23	22	ad2antll 765	. . . . 5 |- ( ( ph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) ) -> P e. ( (Vtx ` G ) ^pm ( 0 ... ( # ` F ) ) ) )
24		snex 4908	. . . . 5 |- { <. ( N + 1 ) , C >. } e. _V
25		unexg 6959	. . . . 5 |- ( ( P e. ( (Vtx ` G ) ^pm ( 0 ... ( # ` F ) ) ) /\ { <. ( N + 1 ) , C >. } e. _V ) -> ( P u. { <. ( N + 1 ) , C >. } ) e. _V )
26	23, 24, 25	sylancl 694	. . . 4 |- ( ( ph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) ) -> ( P u. { <. ( N + 1 ) , C >. } ) e. _V )
27	19, 26	syl5eqel 2705	. . 3 |- ( ( ph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) ) -> Q e. _V )
28	12, 18, 27	3jca 1242	. 2 |- ( ( ph /\ ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) ) ) -> ( S e. _V /\ H e. _V /\ Q e. _V ) )
29	7, 28	mpdan 702	1 |- ( ph -> ( S e. _V /\ H e. _V /\ Q e. _V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   dom cdm 5114   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   ...cfz 12326   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939  Walkscwlks 26492

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-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-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495

This theorem is referenced by:  wlkp1  26578