Theorem wksfval 26505

Description: The set of walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.)

Hypotheses

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

Assertion

Ref	Expression
wksfval	⊢ (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))))})

Distinct variable groups:   𝑓,𝐺,𝑘,𝑝   𝑓,𝐼,𝑝   𝑉,𝑝   𝑓,𝑊

Allowed substitution hints:   𝐼(𝑘)   𝑉(𝑓,𝑘)   𝑊(𝑘,𝑝)

Proof of Theorem wksfval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wlks 26495	. . 3 ⊢ Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))})
2	1	a1i 11	. 2 ⊢ (𝐺 ∈ 𝑊 → Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))}))
3		fveq2 6191	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
4		wksfval.i	. . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐺)
5	3, 4	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
6	5	dmeqd 5326	. . . . . . 7 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼)
7		wrdeq 13327	. . . . . . 7 ⊢ (dom (iEdg‘𝑔) = dom 𝐼 → Word dom (iEdg‘𝑔) = Word dom 𝐼)
8	6, 7	syl 17	. . . . . 6 ⊢ (𝑔 = 𝐺 → Word dom (iEdg‘𝑔) = Word dom 𝐼)
9	8	eleq2d 2687	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑓 ∈ Word dom (iEdg‘𝑔) ↔ 𝑓 ∈ Word dom 𝐼))
10		fveq2 6191	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
11		wksfval.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
12	10, 11	syl6eqr 2674	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
13	12	feq3d 6032	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ↔ 𝑝:(0...(#‘𝑓))⟶𝑉))
14		biidd 252	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)) ↔ (𝑝‘𝑘) = (𝑝‘(𝑘 + 1))))
15	5	fveq1d 6193	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((iEdg‘𝑔)‘(𝑓‘𝑘)) = (𝐼‘(𝑓‘𝑘)))
16	15	eqeq1d 2624	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)} ↔ (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}))
17	15	sseq2d 3633	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ({(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘)) ↔ {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))))
18	14, 16, 17	ifpbi123d 1027	. . . . . 6 ⊢ (𝑔 = 𝐺 → (if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))) ↔ if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘)))))
19	18	ralbidv 2986	. . . . 5 ⊢ (𝑔 = 𝐺 → (∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))) ↔ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘)))))
20	9, 13, 19	3anbi123d 1399	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(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))))))
21	20	opabbidv 4716	. . 3 ⊢ (𝑔 = 𝐺 → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))} = {⟨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))))})
22	21	adantl 482	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑔 = 𝐺) → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))} = {⟨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))))})
23		elex 3212	. 2 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
24		3anass 1042	. . . 4 ⊢ ((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))))))
25	24	opabbii 4717	. . 3 ⊢ {⟨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, 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)))))}
26		fvex 6201	. . . . . . 7 ⊢ (iEdg‘𝐺) ∈ V
27	4, 26	eqeltri 2697	. . . . . 6 ⊢ 𝐼 ∈ V
28	27	dmex 7099	. . . . 5 ⊢ dom 𝐼 ∈ V
29		wrdexg 13315	. . . . 5 ⊢ (dom 𝐼 ∈ V → Word dom 𝐼 ∈ V)
30	28, 29	mp1i 13	. . . 4 ⊢ (𝐺 ∈ 𝑊 → Word dom 𝐼 ∈ V)
31		ovex 6678	. . . . . 6 ⊢ (0...(#‘𝑓)) ∈ V
32		fvex 6201	. . . . . . . 8 ⊢ (Vtx‘𝐺) ∈ V
33	11, 32	eqeltri 2697	. . . . . . 7 ⊢ 𝑉 ∈ V
34	33	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → 𝑉 ∈ V)
35		mapex 7863	. . . . . 6 ⊢ (((0...(#‘𝑓)) ∈ V ∧ 𝑉 ∈ V) → {𝑝 ∣ 𝑝:(0...(#‘𝑓))⟶𝑉} ∈ V)
36	31, 34, 35	sylancr 695	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ 𝑝:(0...(#‘𝑓))⟶𝑉} ∈ V)
37		simpl 473	. . . . . . 7 ⊢ ((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)))) → 𝑝:(0...(#‘𝑓))⟶𝑉)
38	37	ss2abi 3674	. . . . . 6 ⊢ {𝑝 ∣ (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))))} ⊆ {𝑝 ∣ 𝑝:(0...(#‘𝑓))⟶𝑉}
39	38	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ (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))))} ⊆ {𝑝 ∣ 𝑝:(0...(#‘𝑓))⟶𝑉})
40	36, 39	ssexd 4805	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ (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))))} ∈ V)
41	30, 40	opabex3d 7145	. . 3 ⊢ (𝐺 ∈ 𝑊 → {⟨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)))))} ∈ V)
42	25, 41	syl5eqel 2705	. 2 ⊢ (𝐺 ∈ 𝑊 → {⟨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))))} ∈ V)
43	2, 22, 23, 42	fvmptd 6288	1 ⊢ (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))))})

Syntax hints:   → wi 4   ∧ wa 384  if-wif 1012   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  {csn 4177  {cpr 4179  {copab 4712   ↦ cmpt 4729  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:  iswlk  26506  wlkprop  26507  wlkv  26508