Theorem elwspths2on 28304

Description: A simple path of length 2 between two vertices (in a graph) as length 3 string. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 16-Mar-2022.)

Hypothesis

Ref	Expression
elwwlks2on.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
elwspths2on	⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))

Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐺,𝑏   𝑉,𝑏   𝑊,𝑏

Proof of Theorem elwspths2on

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wspthnon 28202	. . . 4 ⊢ (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑊))
2	1	biimpi 215	. . 3 ⊢ (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑊))
3		elwwlks2on.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	elwwlks2on 28303	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2))))
5		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → 𝑊 = ⟨“𝐴𝑏𝐶”⟩)
6		eleq1 2827	. . . . . . . . . . . . . 14 ⊢ (𝑊 = ⟨“𝐴𝑏𝐶”⟩ → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))
7	6	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))
8	5, 7	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))
9	8	ex 412	. . . . . . . . . . 11 ⊢ (𝑊 = ⟨“𝐴𝑏𝐶”⟩ → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
10	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
11	10	com12 32	. . . . . . . . 9 ⊢ (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
12	11	reximdv 3203	. . . . . . . 8 ⊢ (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
13	12	a1i13 27	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑊 → (∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))))
14	13	com24 95	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)) → (∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑊 → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))))
15	4, 14	sylbid 239	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑊 → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))))
16	15	impd 410	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑊) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))))
17	16	com23 86	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑊) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))))
18	2, 17	mpdi 45	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
19	6	biimpar 477	. . . 4 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))
20	19	a1i 11	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))
21	20	rexlimdva 3214	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))
22	18, 21	impbid 211	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541  ∃wex 1785   ∈ wcel 2109  ∃wrex 3066   class class class wbr 5078  ‘cfv 6430  (class class class)co 7268  2c2 12011  ♯chash 14025  ⟨“cs3 14536  Vtxcvtx 27347  UPGraphcupgr 27431  Walkscwlks 27944  SPathsOncspthson 28062   WWalksNOn cwwlksnon 28171   WSPathsNOn cwwspthsnon 28173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-ac2 10203  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-se 5544  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-isom 6439  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-1st 7817  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-1o 8281  df-2o 8282  df-oadd 8285  df-er 8472  df-map 8591  df-pm 8592  df-en 8708  df-dom 8709  df-sdom 8710  df-fin 8711  df-dju 9643  df-card 9681  df-ac 9856  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-nn 11957  df-2 12019  df-3 12020  df-n0 12217  df-xnn0 12289  df-z 12303  df-uz 12565  df-fz 13222  df-fzo 13365  df-hash 14026  df-word 14199  df-concat 14255  df-s1 14282  df-s2 14542  df-s3 14543  df-edg 27399  df-uhgr 27409  df-upgr 27433  df-wlks 27947  df-wwlks 28174  df-wwlksn 28175  df-wwlksnon 28176  df-wspthsnon 28178

This theorem is referenced by:  usgr2wspthon  28309  elwspths2spth  28311