Theorem wksfval 29756

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 29746	. 2 ⊢ 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		fveq2 6863	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
3		wksfval.i	. . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺)
4	2, 3	eqtr4di 2814	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
5	4	dmeqd 5879	. . . . . 6 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼)
6		wrdeq 14546	. . . . . 6 ⊢ (dom (iEdg‘𝑔) = dom 𝐼 → Word dom (iEdg‘𝑔) = Word dom 𝐼)
7	5, 6	syl 17	. . . . 5 ⊢ (𝑔 = 𝐺 → Word dom (iEdg‘𝑔) = Word dom 𝐼)
8	7	eleq2d 2847	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑓 ∈ Word dom (iEdg‘𝑔) ↔ 𝑓 ∈ Word dom 𝐼))
9		fveq2 6863	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
10		wksfval.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
11	9, 10	eqtr4di 2814	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
12	11	feq3d 6672	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ↔ 𝑝:(0...(♯‘𝑓))⟶𝑉))
13	4	fveq1d 6865	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((iEdg‘𝑔)‘(𝑓‘𝑘)) = (𝐼‘(𝑓‘𝑘)))
14	13	eqeq1d 2763	. . . . . 6 ⊢ (𝑔 = 𝐺 → (((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)} ↔ (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}))
15	13	sseq2d 3968	. . . . . 6 ⊢ (𝑔 = 𝐺 → ({(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘)) ↔ {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))))
16	14, 15	ifpbi23d 1090	. . . . 5 ⊢ (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)))))
17	16	ralbidv 3184	. . . 4 ⊢ (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)))))
18	8, 12, 17	3anbi123d 1456	. . 3 ⊢ (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))))))
19	18	opabbidv 5165	. 2 ⊢ (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))))})
20		elex 3474	. 2 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
21		3anass 1105	. . . 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))))))
22	21	opabbii 5166	. . 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)))))}
23	3	fvexi 6877	. . . . . 6 ⊢ 𝐼 ∈ V
24	23	dmex 7886	. . . . 5 ⊢ dom 𝐼 ∈ V
25		wrdexg 14534	. . . . 5 ⊢ (dom 𝐼 ∈ V → Word dom 𝐼 ∈ V)
26	24, 25	mp1i 13	. . . 4 ⊢ (𝐺 ∈ 𝑊 → Word dom 𝐼 ∈ V)
27		ovex 7425	. . . . . 6 ⊢ (0...(♯‘𝑓)) ∈ V
28	10	fvexi 6877	. . . . . . 7 ⊢ 𝑉 ∈ V
29	28	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → 𝑉 ∈ V)
30		mapex 7917	. . . . . 6 ⊢ (((0...(♯‘𝑓)) ∈ V ∧ 𝑉 ∈ V) → {𝑝 ∣ 𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
31	27, 29, 30	sylancr 596	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ 𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
32		simpl 486	. . . . . . 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)
33	32	ss2abi 4019	. . . . . 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}
34	33	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})
35	31, 34	ssexd 5279	. . . 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)
36	26, 35	opabex3d 7942	. . 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)
37	22, 36	eqeltrid 2865	. 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)
38	1, 19, 20, 37	fvmptd3 6995	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 399  if-wif 1073   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075  Vcvv 3453   ⊆ wss 3904  {csn 4581  {cpr 4583  {copab 5161  dom cdm 5645  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  0cc0 11070  1c1 11071   + caddc 11073  ...cfz 13509  ..^cfzo 13656  ♯chash 14340  Word cword 14523  Vtxcvtx 29143  iEdgciedg 29144  Walkscwlks 29743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ifp 1074  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-n0 12479  df-z 12566  df-uz 12837  df-fz 13510  df-fzo 13657  df-hash 14341  df-word 14524  df-wlks 29746

This theorem is referenced by:  iswlk  29757  wlkprop  29758  wlkv  29759