numclwlk1lem2fo - Metamath Proof Explorer

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Theorem numclwlk1lem2fo 27228

Description:

is an onto function. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 29-May-2021.)

**Hypotheses**
Ref	Expression
extwwlkfab.v	Vtx
extwwlkfab.f	ClWWalksN
extwwlkfab.c	ClWWalksN $/\$
numclwwlk.t	substr

**Assertion**
Ref	Expression
numclwlk1lem2fo	USGraph $/\$ $/\$ NeighbVtx

Distinct variable groups:

Allowed substitution hints:

(

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(

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(

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Proof of Theorem numclwlk1lem2fo Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		extwwlkfab.v	. . 3 Vtx
2		extwwlkfab.f	. . 3 ClWWalksN
3		extwwlkfab.c	. . 3 ClWWalksN $/\$
4		numclwwlk.t	. . 3 substr
5	1, 2, 3, 4	numclwlk1lem2f 27225	. 2 USGraph $/\$ $/\$ NeighbVtx
6		elxp 5131	. . . . 5 NeighbVtx $/\$ $/\$ NeighbVtx
7	1, 2, 3	numclwlk1lem2foa 27224	. . . . . . . . . . 11 USGraph $/\$ $/\$ $/\$ NeighbVtx ++ ++
8	7	com12 32	. . . . . . . . . 10 $/\$ NeighbVtx USGraph $/\$ $/\$ ++ ++
9	8	adantl 482	. . . . . . . . 9 $/\$ $/\$ NeighbVtx USGraph $/\$ $/\$ ++ ++
10	9	imp 445	. . . . . . . 8 $/\$ $/\$ NeighbVtx $/\$ USGraph $/\$ $/\$ ++ ++
11		simpl 473	. . . . . . . . 9 ++ ++ $/\$ $/\$ $/\$ NeighbVtx $/\$ USGraph $/\$ $/\$ ++ ++
12		fveq2 6191	. . . . . . . . . . 11 ++ ++ ++ ++
13	12	eqeq2d 2632	. . . . . . . . . 10 ++ ++ ++ ++
14	1, 2, 3, 4	numclwlk1lem2fv 27226	. . . . . . . . . . . 12 ++ ++ ++ ++ ++ ++ substr ++ ++
15	14	adantr 481	. . . . . . . . . . 11 ++ ++ $/\$ $/\$ $/\$ NeighbVtx $/\$ USGraph $/\$ $/\$ ++ ++ ++ ++ substr ++ ++
16	15	eqeq2d 2632	. . . . . . . . . 10 ++ ++ $/\$ $/\$ $/\$ NeighbVtx $/\$ USGraph $/\$ $/\$ ++ ++ ++ ++ substr ++ ++
17	13, 16	sylan9bbr 737	. . . . . . . . 9 ++ ++ $/\$ $/\$ $/\$ NeighbVtx $/\$ USGraph $/\$ $/\$ $/\$ ++ ++ ++ ++ substr ++ ++
18		simprll 802	. . . . . . . . . 10 ++ ++ $/\$ $/\$ $/\$ NeighbVtx $/\$ USGraph $/\$ $/\$
19	1	nbgrisvtx 26255	. . . . . . . . . . . . . . . . . 18 USGraph $/\$ NeighbVtx
20	19	ex 450	. . . . . . . . . . . . . . . . 17 USGraph NeighbVtx
21	20	3ad2ant1 1082	. . . . . . . . . . . . . . . 16 USGraph $/\$ $/\$ NeighbVtx
22		simp1 1061	. . . . . . . . . . . . . . . . . . . 20 USGraph $/\$ $/\$ USGraph
23		uz3m2nn 11731	. . . . . . . . . . . . . . . . . . . . 21
24	23	3ad2ant3 1084	. . . . . . . . . . . . . . . . . . . 20 USGraph $/\$ $/\$
25		simp2 1062	. . . . . . . . . . . . . . . . . . . 20 USGraph $/\$ $/\$
26		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 Edg Edg
27	2, 1, 26	numclwwlkovfel2 27216	. . . . . . . . . . . . . . . . . . . 20 USGraph $/\$ $/\$ Word $/\$ ..^ Edg $/\$ lastS Edg $/\$ $/\$
28	22, 24, 25, 27	syl3anc 1326	. . . . . . . . . . . . . . . . . . 19 USGraph $/\$ $/\$ Word $/\$ ..^ Edg $/\$ lastS Edg $/\$ $/\$
29		df-3an 1039	. . . . . . . . . . . . . . . . . . 19 Word $/\$ ..^ Edg $/\$ lastS Edg $/\$ $/\$ Word $/\$ ..^ Edg $/\$ lastS Edg $/\$ $/\$
30	28, 29	syl6bb 276	. . . . . . . . . . . . . . . . . 18 USGraph $/\$ $/\$ Word $/\$ ..^ Edg $/\$ lastS Edg $/\$ $/\$
31		simplll 798	. . . . . . . . . . . . . . . . . . . . . . . . . 26 Word $/\$ $/\$ $/\$ $/\$ Word
32		s1cl 13382	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Word
33	32	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 $/\$ Word
34	33	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 Word $/\$ $/\$ $/\$ Word
35	34	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 Word $/\$ $/\$ $/\$ $/\$ Word
36		s1cl 13382	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 Word
37	36	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 Word $/\$ $/\$ $/\$ $/\$ Word
38		ccatass 13371	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 Word $/\$ Word $/\$ Word ++ ++ ++ ++
39	38	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . 26 Word $/\$ Word $/\$ Word ++ ++ substr ++ ++ substr
40	31, 35, 37, 39	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . 25 Word $/\$ $/\$ $/\$ $/\$ ++ ++ substr ++ ++ substr
41		ccatcl 13359	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 Word $/\$ Word ++ Word
42	34, 36, 41	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . . 26 Word $/\$ $/\$ $/\$ $/\$ ++ Word
43		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Word $/\$
44	43	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Word $/\$
45	44	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 Word $/\$ $/\$ $/\$
46	45	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 Word $/\$ $/\$ $/\$ $/\$
47		swrdccatid 13497	. . . . . . . . . . . . . . . . . . . . . . . . . 26 Word $/\$ ++ Word $/\$ ++ ++ substr
48	31, 42, 46, 47	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . 25 Word $/\$ $/\$ $/\$ $/\$ ++ ++ substr
49	40, 48	eqtr2d 2657	. . . . . . . . . . . . . . . . . . . . . . . 24 Word $/\$ $/\$ $/\$ $/\$ ++ ++ substr
50		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ++ ++
51		lsw 13351	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ++ ++ lastS ++ ++ ++ ++ ++ ++
52	50, 51	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 lastS ++ ++ ++ ++ ++ ++
53		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 Word $/\$ Word
54		ccatcl 13359	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 Word $/\$ Word ++ Word
55	53, 33, 54	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . . 26 Word $/\$ $/\$ $/\$ ++ Word
56		lswccats1 13411	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ++ Word $/\$ lastS ++ ++
57	55, 56	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . 25 Word $/\$ $/\$ $/\$ $/\$ lastS ++ ++
58		ccatlen 13360	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ++ Word $/\$ Word ++ ++ ++
59	55, 36, 58	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Word $/\$ $/\$ $/\$ $/\$ ++ ++ ++
60	53, 33	anim12i 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Word $/\$ $/\$ $/\$ Word $/\$ Word
61	60	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Word $/\$ $/\$ $/\$ $/\$ Word $/\$ Word
62		ccatlen 13360	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Word $/\$ Word ++
63	61, 62	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Word $/\$ $/\$ $/\$ $/\$ ++
64		s1len 13385	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
65	64	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Word $/\$ $/\$ $/\$ $/\$
66	63, 65	oveq12d 6668	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Word $/\$ $/\$ $/\$ $/\$ ++
67		s1len 13385	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
68	67	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 $/\$
69	43, 68	oveqan12d 6669	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Word $/\$ $/\$ $/\$
70	69	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Word $/\$ $/\$ $/\$
71		eluzelcn 11699	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
72		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
73		2cnd 11093	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
74	72, 73	subcld 10392	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
75		1cnd 10056	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
76	74, 75, 75	addassd 10062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
77		1p1e2 11134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
78	77	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
79	78	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
80	76, 79	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
81	71, 80	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
82		2cnd 11093	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
83	71, 82	npcand 10396	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
84	81, 83	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
85	84	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 $/\$
86	85	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Word $/\$ $/\$ $/\$
87	70, 86	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Word $/\$ $/\$ $/\$
88	87	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Word $/\$ $/\$ $/\$ $/\$
89	59, 66, 88	3eqtrd 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 Word $/\$ $/\$ $/\$ $/\$ ++ ++
90	89	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . 26 Word $/\$ $/\$ $/\$ $/\$ ++ ++
91	90	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . 25 Word $/\$ $/\$ $/\$ $/\$ ++ ++ ++ ++ ++ ++
92	52, 57, 91	3eqtr3a 2680	. . . . . . . . . . . . . . . . . . . . . . . 24 Word $/\$ $/\$ $/\$ $/\$ ++ ++
93	49, 92	opeq12d 4410	. . . . . . . . . . . . . . . . . . . . . . 23 Word $/\$ $/\$ $/\$ $/\$ ++ ++ substr ++ ++
94	93	exp31 630	. . . . . . . . . . . . . . . . . . . . . 22 Word $/\$ $/\$ ++ ++ substr ++ ++
95	94	3ad2antl1 1223	. . . . . . . . . . . . . . . . . . . . 21 Word $/\$ ..^ Edg $/\$ lastS Edg $/\$ $/\$ ++ ++ substr ++ ++
96	95	adantr 481	. . . . . . . . . . . . . . . . . . . 20 Word $/\$ ..^ Edg $/\$ lastS Edg $/\$ $/\$ $/\$ ++ ++ substr ++ ++
97	96	com12 32	. . . . . . . . . . . . . . . . . . 19 $/\$ Word $/\$ ..^ Edg $/\$ lastS Edg $/\$ $/\$ ++ ++ substr ++ ++
98	97	3adant1 1079	. . . . . . . . . . . . . . . . . 18 USGraph $/\$ $/\$ Word $/\$ ..^ Edg $/\$ lastS Edg $/\$ $/\$ ++ ++ substr ++ ++
99	30, 98	sylbid 230	. . . . . . . . . . . . . . . . 17 USGraph $/\$ $/\$ ++ ++ substr ++ ++
100	99	com23 86	. . . . . . . . . . . . . . . 16 USGraph $/\$ $/\$ ++ ++ substr ++ ++
101	21, 100	syld 47	. . . . . . . . . . . . . . 15 USGraph $/\$ $/\$ NeighbVtx ++ ++ substr ++ ++
102	101	com13 88	. . . . . . . . . . . . . 14 NeighbVtx USGraph $/\$ $/\$ ++ ++ substr ++ ++
103	102	imp 445	. . . . . . . . . . . . 13 $/\$ NeighbVtx USGraph $/\$ $/\$ ++ ++ substr ++ ++
104	103	adantl 482	. . . . . . . . . . . 12 $/\$ $/\$ NeighbVtx USGraph $/\$ $/\$ ++ ++ substr ++ ++
105	104	imp 445	. . . . . . . . . . 11 $/\$ $/\$ NeighbVtx $/\$ USGraph $/\$ $/\$ ++ ++ substr ++ ++
106	105	adantl 482	. . . . . . . . . 10 ++ ++ $/\$ $/\$ $/\$ NeighbVtx $/\$ USGraph $/\$ $/\$ ++ ++ substr ++ ++
107	18, 106	eqtrd 2656	. . . . . . . . 9 ++ ++ $/\$ $/\$ $/\$ NeighbVtx $/\$ USGraph $/\$ $/\$ ++ ++ substr ++ ++
108	11, 17, 107	rspcedvd 3317	. . . . . . . 8 ++ ++ $/\$ $/\$ $/\$ NeighbVtx $/\$ USGraph $/\$ $/\$
109	10, 108	mpancom 703	. . . . . . 7 $/\$ $/\$ NeighbVtx $/\$ USGraph $/\$ $/\$
110	109	ex 450	. . . . . 6 $/\$ $/\$ NeighbVtx USGraph $/\$ $/\$
111	110	exlimivv 1860	. . . . 5 $/\$ $/\$ NeighbVtx USGraph $/\$ $/\$
112	6, 111	sylbi 207	. . . 4 NeighbVtx USGraph $/\$ $/\$
113	112	impcom 446	. . 3 USGraph $/\$ $/\$ $/\$ NeighbVtx
114	113	ralrimiva 2966	. 2 USGraph $/\$ $/\$ NeighbVtx
115		dffo3 6374	. 2 NeighbVtx NeighbVtx $/\$ NeighbVtx
116	5, 114, 115	sylanbrc 698	1 USGraph $/\$ $/\$ NeighbVtx

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wex 1704

wcel 1990

wral 2912

wrex 2913

crab 2916

cvv 3200

cpr 4179

cop 4183

cmpt 4729

cxp 5112

wf 5884

wfo 5886

cfv 5888 (class class class)co 6650

cmpt2 6652

cc 9934

cc0 9936

c1 9937

caddc 9939

cmin 10266

cn 11020

c2 11070

c3 11071

cuz 11687 ..^cfzo 12465

chash 13117 Word cword 13291 lastS clsw 13292 ++ cconcat 13293

cs1 13294 substr csubstr 13295 Vtxcvtx 25874 Edgcedg 25939 USGraph cusgr 26044 NeighbVtx cnbgr 26224 ClWWalksN cclwwlksn 26876

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-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-edg 25940 df-upgr 25977 df-umgr 25978 df-usgr 26046 df-nbgr 26228 df-wwlks 26722 df-wwlksn 26723 df-clwwlks 26877 df-clwwlksn 26878

This theorem is referenced by: numclwlk1lem2f1o 27229

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