Theorem upwlksfval 48297

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 48296	. 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 6831	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
3		upwlksfval.i	. . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺)
4	2, 3	eqtr4di 2786	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
5	4	dmeqd 5851	. . . . . 6 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼)
6		wrdeq 14450	. . . . . 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 2819	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑓 ∈ Word dom (iEdg‘𝑔) ↔ 𝑓 ∈ Word dom 𝐼))
9		fveq2 6831	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
10		upwlksfval.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
11	9, 10	eqtr4di 2786	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
12	11	feq3d 6644	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ↔ 𝑝:(0...(♯‘𝑓))⟶𝑉))
13	4	fveq1d 6833	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((iEdg‘𝑔)‘(𝑓‘𝑘)) = (𝐼‘(𝑓‘𝑘)))
14	13	eqeq1d 2735	. . . . 5 ⊢ (𝑔 = 𝐺 → (((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}))
15	14	ralbidv 3156	. . . 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 1438	. . 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 5161	. 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 3458	. 2 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
19		3anass 1094	. . . 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 5162	. . 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 6845	. . . . . 6 ⊢ 𝐼 ∈ V
22	21	dmex 7848	. . . . 5 ⊢ dom 𝐼 ∈ V
23		wrdexg 14438	. . . . 5 ⊢ (dom 𝐼 ∈ V → Word dom 𝐼 ∈ V)
24	22, 23	mp1i 13	. . . 4 ⊢ (𝐺 ∈ 𝑊 → Word dom 𝐼 ∈ V)
25		ovex 7388	. . . . . 6 ⊢ (0...(♯‘𝑓)) ∈ V
26	10	fvexi 6845	. . . . . . 7 ⊢ 𝑉 ∈ V
27	26	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → 𝑉 ∈ V)
28		mapex 7880	. . . . . 6 ⊢ (((0...(♯‘𝑓)) ∈ V ∧ 𝑉 ∈ V) → {𝑝 ∣ 𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
29	25, 27, 28	sylancr 587	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ 𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
30		simpl 482	. . . . . . 7 ⊢ ((𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}) → 𝑝:(0...(♯‘𝑓))⟶𝑉)
31	30	ss2abi 4015	. . . . . 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 5266	. . . 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 7906	. . 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 2837	. 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 6961	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))})})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {cab 2711  ∀wral 3048  Vcvv 3437   ⊆ wss 3898  {cpr 4579  {copab 5157  dom cdm 5621  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355  0cc0 11017  1c1 11018   + caddc 11020  ...cfz 13414  ..^cfzo 13561  ♯chash 14244  Word cword 14427  Vtxcvtx 28995  iEdgciedg 28996  UPWalkscupwlks 48295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-map 8761  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9843  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-n0 12393  df-z 12480  df-uz 12743  df-fz 13415  df-fzo 13562  df-hash 14245  df-word 14428  df-upwlks 48296

This theorem is referenced by:  isupwlk  48298  isupwlkg  48299  upwlkbprop  48300