Theorem iswlk 26506

Description: Properties of a pair of functions to be a walk. (Contributed by AV, 30-Dec-2020.)

Hypotheses

Ref	Expression
wksfval.v	⊢ 𝑉 = (Vtx‘𝐺)
wksfval.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
iswlk	⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))

Distinct variable groups:   𝑘,𝐺   𝑘,𝐹   𝑃,𝑘

Allowed substitution hints:   𝑈(𝑘)   𝐼(𝑘)   𝑉(𝑘)   𝑊(𝑘)   𝑍(𝑘)

Proof of Theorem iswlk

Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 4654	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (Walks‘𝐺))
2		wksfval.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
3		wksfval.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
4	2, 3	wksfval 26505	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (Walks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))))})
5	4	3ad2ant1 1082	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (Walks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))))})
6	5	eleq2d 2687	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (⟨𝐹, 𝑃⟩ ∈ (Walks‘𝐺) ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))))}))
7	1, 6	syl5bb 272	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))))}))
8		eleq1 2689	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ Word dom 𝐼 ↔ 𝐹 ∈ Word dom 𝐼))
9	8	adantr 481	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓 ∈ Word dom 𝐼 ↔ 𝐹 ∈ Word dom 𝐼))
10		simpr 477	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
11		fveq2 6191	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (#‘𝑓) = (#‘𝐹))
12	11	oveq2d 6666	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (0...(#‘𝑓)) = (0...(#‘𝐹)))
13	12	adantr 481	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (0...(#‘𝑓)) = (0...(#‘𝐹)))
14	10, 13	feq12d 6033	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑝:(0...(#‘𝑓))⟶𝑉 ↔ 𝑃:(0...(#‘𝐹))⟶𝑉))
15	11	oveq2d 6666	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (0..^(#‘𝑓)) = (0..^(#‘𝐹)))
16	15	adantr 481	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (0..^(#‘𝑓)) = (0..^(#‘𝐹)))
17		fveq1 6190	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑘) = (𝑃‘𝑘))
18		fveq1 6190	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
19	17, 18	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)) ↔ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))))
20	19	adantl 482	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)) ↔ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))))
21		fveq1 6190	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑘) = (𝐹‘𝑘))
22	21	fveq2d 6195	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝐼‘(𝑓‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
23	17	sneqd 4189	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → {(𝑝‘𝑘)} = {(𝑃‘𝑘)})
24	22, 23	eqeqan12d 2638	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)} ↔ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}))
25	17, 18	preq12d 4276	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
26	25	adantl 482	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
27	22	adantr 481	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝐼‘(𝑓‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
28	26, 27	sseq12d 3634	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ({(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘)) ↔ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
29	20, 24, 28	ifpbi123d 1027	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))) ↔ if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))
30	16, 29	raleqbidv 3152	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))) ↔ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))
31	9, 14, 30	3anbi123d 1399	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘)))) ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
32	31	opelopabga 4988	. . 3 ⊢ ((𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))))} ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
33	32	3adant1 1079	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))))} ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
34	7, 33	bitrd 268	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  if-wif 1012   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  {csn 4177  {cpr 4179  ⟨cop 4183   class class class wbr 4653  {copab 4712  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  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:  wlkprop  26507  iswlkg  26509  wlkvtxeledg  26519  wlk1walk  26535  wlkres  26567  redwlk  26569  wlkp1  26578  wlkd  26583  lfgrwlkprop  26584  crctcshwlkn0  26713  upwlkwlk  41720  upgrwlkupwlk  41721