Theorem upwlksfval 49202

Description: The set of simple walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 28-Dec-2020.)

Hypotheses

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

Assertion

Ref	Expression
upwlksfval	⊢ (𝐺 ∈ 𝑊 → (UPWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})

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 upwlksfval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-upwlks 49201	. 2 ⊢ UPWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})
2		fveq2 6883	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
3		upwlksfval.i	. . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺)
4	2, 3	eqtr4di 2814	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
5	4	dmeqd 5887	. . . . . 6 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼)
6		wrdeq 14674	. . . . . 6 ⊢ (dom (iEdg‘𝑔) = dom 𝐼 → Word dom (iEdg‘𝑔) = Word dom 𝐼)
7	5, 6	syl 18	. . . . 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 6883	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
10		upwlksfval.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
11	9, 10	eqtr4di 2814	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
12	11	feq3d 6692	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ↔ 𝑝:(0...(♯‘𝑓))⟶𝑉))
13	4	fveq1d 6885	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((iEdg‘𝑔)‘(𝑓‘𝑘)) = (𝐼‘(𝑓‘𝑘)))
14	13	eqeq1d 2763	. . . . 5 ⊢ (𝑔 = 𝐺 → (((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}))
15	14	ralbidv 3186	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}))
16	8, 12, 15	3anbi123d 1464	. . 3 ⊢ (𝑔 = 𝐺 → ((𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}) ↔ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})))
17	16	opabbidv 5171	. 2 ⊢ (𝑔 = 𝐺 → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})
18		elex 3472	. 2 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
19		3anass 1111	. . . 4 ⊢ ((𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}) ↔ (𝑓 ∈ Word dom 𝐼 ∧ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})))
20	19	opabbii 5172	. . 3 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}))}
21	3	fvexi 6897	. . . . . 6 ⊢ 𝐼 ∈ V
22	21	dmex 7919	. . . . 5 ⊢ dom 𝐼 ∈ V
23		wrdexg 14662	. . . . 5 ⊢ (dom 𝐼 ∈ V → Word dom 𝐼 ∈ V)
24	22, 23	mp1i 14	. . . 4 ⊢ (𝐺 ∈ 𝑊 → Word dom 𝐼 ∈ V)
25		ovex 7451	. . . . . 6 ⊢ (0...(♯‘𝑓)) ∈ V
26	10	fvexi 6897	. . . . . . 7 ⊢ 𝑉 ∈ V
27	26	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → 𝑉 ∈ V)
28		mapex 7950	. . . . . 6 ⊢ (((0...(♯‘𝑓)) ∈ V ∧ 𝑉 ∈ V) → {𝑝 ∣ 𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
29	25, 27, 28	sylancr 599	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ 𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
30		simpl 488	. . . . . . 7 ⊢ ((𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}) → 𝑝:(0...(♯‘𝑓))⟶𝑉)
31	30	ss2abi 4014	. . . . . 6 ⊢ {𝑝 ∣ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} ⊆ {𝑝 ∣ 𝑝:(0...(♯‘𝑓))⟶𝑉}
32	31	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} ⊆ {𝑝 ∣ 𝑝:(0...(♯‘𝑓))⟶𝑉})
33	29, 32	ssexd 5286	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} ∈ V)
34	24, 33	opabex3d 7975	. . 3 ⊢ (𝐺 ∈ 𝑊 → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}))} ∈ V)
35	20, 34	eqeltrid 2865	. 2 ⊢ (𝐺 ∈ 𝑊 → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} ∈ V)
36	1, 17, 18, 35	fvmptd3 7015	1 ⊢ (𝐺 ∈ 𝑊 → (UPWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  Vcvv 3451   ⊆ wss 3899  {cpr 4586  {copab 5167  dom cdm 5651  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  0cc0 11193  1c1 11194   + caddc 11196  ...cfz 13632  ..^cfzo 13781  ♯chash 14467  Word cword 14651  Vtxcvtx 29567  iEdgciedg 29568  UPWalkscupwlks 49200

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-n0 12600  df-z 12687  df-uz 12959  df-fz 13633  df-fzo 13782  df-hash 14468  df-word 14652  df-upwlks 49201

This theorem is used by:  isupwlk  49203  isupwlkg  49204  upwlkbprop  49205