Theorem upwlksfval 47169

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 47168	. 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 6891	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
3		upwlksfval.i	. . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺)
4	2, 3	eqtr4di 2785	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
5	4	dmeqd 5902	. . . . . 6 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼)
6		wrdeq 14512	. . . . . 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 2814	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑓 ∈ Word dom (iEdg‘𝑔) ↔ 𝑓 ∈ Word dom 𝐼))
9		fveq2 6891	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
10		upwlksfval.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
11	9, 10	eqtr4di 2785	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
12	11	feq3d 6703	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ↔ 𝑝:(0...(♯‘𝑓))⟶𝑉))
13	4	fveq1d 6893	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((iEdg‘𝑔)‘(𝑓‘𝑘)) = (𝐼‘(𝑓‘𝑘)))
14	13	eqeq1d 2729	. . . . 5 ⊢ (𝑔 = 𝐺 → (((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}))
15	14	ralbidv 3172	. . . 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 1433	. . 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 5208	. 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 3488	. 2 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
19		3anass 1093	. . . 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 5209	. . 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 6905	. . . . . 6 ⊢ 𝐼 ∈ V
22	21	dmex 7911	. . . . 5 ⊢ dom 𝐼 ∈ V
23		wrdexg 14500	. . . . 5 ⊢ (dom 𝐼 ∈ V → Word dom 𝐼 ∈ V)
24	22, 23	mp1i 13	. . . 4 ⊢ (𝐺 ∈ 𝑊 → Word dom 𝐼 ∈ V)
25		ovex 7447	. . . . . 6 ⊢ (0...(♯‘𝑓)) ∈ V
26	10	fvexi 6905	. . . . . . 7 ⊢ 𝑉 ∈ V
27	26	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼) → 𝑉 ∈ V)
28		mapex 8844	. . . . . 6 ⊢ (((0...(♯‘𝑓)) ∈ V ∧ 𝑉 ∈ V) → {𝑝 ∣ 𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
29	25, 27, 28	sylancr 586	. . . . 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 4059	. . . . . 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 5318	. . . 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 7963	. . 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 2832	. 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 7022	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 1085   = wceq 1534   ∈ wcel 2099  {cab 2704  ∀wral 3056  Vcvv 3469   ⊆ wss 3944  {cpr 4626  {copab 5204  dom cdm 5672  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414  0cc0 11132  1c1 11133   + caddc 11135  ...cfz 13510  ..^cfzo 13653  ♯chash 14315  Word cword 14490  Vtxcvtx 28802  iEdgciedg 28803  UPWalkscupwlks 47167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-card 9956  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-nn 12237  df-n0 12497  df-z 12583  df-uz 12847  df-fz 13511  df-fzo 13654  df-hash 14316  df-word 14491  df-upwlks 47168

This theorem is referenced by:  isupwlk  47170  isupwlkg  47171  upwlkbprop  47172