Theorem iswlk 29757

Description: Properties of a pair of functions to be/represent 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 5100	. . 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 29756	. . . . 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 1145	. . . 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 2847	. . 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	bitrid 285	. 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 2849	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ Word dom 𝐼 ↔ 𝐹 ∈ Word dom 𝐼))
9	8	adantr 484	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓 ∈ Word dom 𝐼 ↔ 𝐹 ∈ Word dom 𝐼))
10		simpr 488	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
11		fveq2 6863	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
12	11	oveq2d 7408	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (0...(♯‘𝑓)) = (0...(♯‘𝐹)))
13	12	adantr 484	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (0...(♯‘𝑓)) = (0...(♯‘𝐹)))
14	10, 13	feq12d 6675	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑝:(0...(♯‘𝑓))⟶𝑉 ↔ 𝑃:(0...(♯‘𝐹))⟶𝑉))
15	11	oveq2d 7408	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
16	15	adantr 484	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
17		fveq1 6862	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑘) = (𝑃‘𝑘))
18		fveq1 6862	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
19	17, 18	eqeq12d 2777	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)) ↔ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))))
20	19	adantl 485	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)) ↔ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))))
21		fveq1 6862	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑘) = (𝐹‘𝑘))
22	21	fveq2d 6867	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝐼‘(𝑓‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
23	17	sneqd 4593	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → {(𝑝‘𝑘)} = {(𝑃‘𝑘)})
24	22, 23	eqeqan12d 2775	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)} ↔ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}))
25	17, 18	preq12d 4699	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
26	25	adantl 485	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
27	22	adantr 484	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝐼‘(𝑓‘𝑘)) = (𝐼‘(𝐹‘𝑘)))
28	26, 27	sseq12d 3969	. . . . . . 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 1089	. . . . . 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 3335	. . . . 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 1456	. . . 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 5502	. . 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 1142	. 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 281	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 208   ∧ wa 399  if-wif 1073   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ⊆ wss 3904  {csn 4581  {cpr 4583  ⟨cop 4587   class class class wbr 5099  {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:  wlkprop  29758  iswlkg  29760  wlkvtxeledg  29770  wlk1walk  29785  redwlk  29817  wlkp1  29826  wlkd  29831  lfgrwlkprop  29832  crctcshwlkn0  29967  upwlkwlk  48725  upgrwlkupwlk  48726