Theorem upgr2wlk 27458

Description: Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 28-Oct-2021.)

Hypotheses

Ref	Expression
upgr2wlk.v	⊢ 𝑉 = (Vtx‘𝐺)
upgr2wlk.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
upgr2wlk	⊢ (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))

Proof of Theorem upgr2wlk

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgr2wlk.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		upgr2wlk.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	upgriswlk 27430	. . 3 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
4	3	anbi1d 632	. 2 ⊢ (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (♯‘𝐹) = 2)))
5		iswrdb 13863	. . . . . . . . 9 ⊢ (𝐹 ∈ Word dom 𝐼 ↔ 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
6		oveq2 7143	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
7	6	feq2d 6473	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 2 → (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ↔ 𝐹:(0..^2)⟶dom 𝐼))
8	5, 7	syl5bb 286	. . . . . . . 8 ⊢ ((♯‘𝐹) = 2 → (𝐹 ∈ Word dom 𝐼 ↔ 𝐹:(0..^2)⟶dom 𝐼))
9		oveq2 7143	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 2 → (0...(♯‘𝐹)) = (0...2))
10	9	feq2d 6473	. . . . . . . 8 ⊢ ((♯‘𝐹) = 2 → (𝑃:(0...(♯‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
11		fzo0to2pr 13117	. . . . . . . . . . 11 ⊢ (0..^2) = {0, 1}
12	6, 11	eqtrdi 2849	. . . . . . . . . 10 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
13	12	raleqdv 3364	. . . . . . . . 9 ⊢ ((♯‘𝐹) = 2 → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
14		2wlklem 27457	. . . . . . . . 9 ⊢ (∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
15	13, 14	syl6bb 290	. . . . . . . 8 ⊢ ((♯‘𝐹) = 2 → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
16	8, 10, 15	3anbi123d 1433	. . . . . . 7 ⊢ ((♯‘𝐹) = 2 → ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
17	16	adantl 485	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (♯‘𝐹) = 2) → ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
18		3anass 1092	. . . . . 6 ⊢ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
19	17, 18	syl6bb 290	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (♯‘𝐹) = 2) → ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))))
20	19	ex 416	. . . 4 ⊢ (𝐺 ∈ UPGraph → ((♯‘𝐹) = 2 → ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))))
21	20	pm5.32rd 581	. . 3 ⊢ (𝐺 ∈ UPGraph → (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (♯‘𝐹) = 2) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ (♯‘𝐹) = 2)))
22		3anass 1092	. . . 4 ⊢ (((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
23		an32 645	. . . 4 ⊢ (((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ (♯‘𝐹) = 2))
24	22, 23	bitri 278	. . 3 ⊢ (((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ (♯‘𝐹) = 2))
25	21, 24	syl6bbr 292	. 2 ⊢ (𝐺 ∈ UPGraph → (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (♯‘𝐹) = 2) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
26		2nn0 11902	. . . . . . 7 ⊢ 2 ∈ ℕ0
27		fnfzo0hash 13804	. . . . . . 7 ⊢ ((2 ∈ ℕ0 ∧ 𝐹:(0..^2)⟶dom 𝐼) → (♯‘𝐹) = 2)
28	26, 27	mpan 689	. . . . . 6 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → (♯‘𝐹) = 2)
29	28	pm4.71i 563	. . . . 5 ⊢ (𝐹:(0..^2)⟶dom 𝐼 ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2))
30	29	bicomi 227	. . . 4 ⊢ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ↔ 𝐹:(0..^2)⟶dom 𝐼)
31	30	a1i 11	. . 3 ⊢ (𝐺 ∈ UPGraph → ((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ↔ 𝐹:(0..^2)⟶dom 𝐼))
32	31	3anbi1d 1437	. 2 ⊢ (𝐺 ∈ UPGraph → (((𝐹:(0..^2)⟶dom 𝐼 ∧ (♯‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
33	4, 25, 32	3bitrd 308	1 ⊢ (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {cpr 4527   class class class wbr 5030  dom cdm 5519  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  0cc0 10526  1c1 10527   + caddc 10529  2c2 11680  ℕ₀cn0 11885  ...cfz 12885  ..^cfzo 13028  ♯chash 13686  Word cword 13857  Vtxcvtx 26789  iEdgciedg 26790  UPGraphcupgr 26873  Walkscwlks 27386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-edg 26841  df-uhgr 26851  df-upgr 26875  df-wlks 27389

This theorem is referenced by:  umgrwwlks2on  27743