Theorem elwwlks2ons3 28221

Description: For each walk of length 2 between two vertices, there is a third vertex in the middle of the walk. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)

Hypothesis

Ref	Expression
wwlks2onv.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
elwwlks2ons3	⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))

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

Proof of Theorem elwwlks2ons3

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
2		wwlks2onv.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	elwwlks2ons3im 28220	. . . 4 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
4		anass 468	. . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) ↔ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
5	1, 3, 4	sylanbrc 582	. . 3 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉))
6		simpr 484	. . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊‘1) ∈ 𝑉)
7		s3eq2 14511	. . . . . 6 ⊢ (𝑏 = (𝑊‘1) → ⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩)
8		eqeq2 2750	. . . . . . 7 ⊢ (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ↔ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩))
9		eleq1 2826	. . . . . . 7 ⊢ (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
10	8, 9	anbi12d 630	. . . . . 6 ⊢ (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
11	7, 10	syl 17	. . . . 5 ⊢ (𝑏 = (𝑊‘1) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
12	11	adantl 481	. . . 4 ⊢ ((((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) ∧ 𝑏 = (𝑊‘1)) → ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
13		simpr 484	. . . . . 6 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩)
14		eleq1 2826	. . . . . . 7 ⊢ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
15	14	biimpac 478	. . . . . 6 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
16	13, 15	jca 511	. . . . 5 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
17	16	adantr 480	. . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
18	6, 12, 17	rspcedvd 3555	. . 3 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
19	5, 18	syl 17	. 2 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
20		eleq1 2826	. . . . 5 ⊢ (⟨“𝐴𝑏𝐶”⟩ = 𝑊 → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
21	20	eqcoms 2746	. . . 4 ⊢ (𝑊 = ⟨“𝐴𝑏𝐶”⟩ → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
22	21	biimpa 476	. . 3 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
23	22	rexlimivw 3210	. 2 ⊢ (∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
24	19, 23	impbii 208	1 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  ‘cfv 6418  (class class class)co 7255  1c1 10803  2c2 11958  ⟨“cs3 14483  Vtxcvtx 27269   WWalksNOn cwwlksnon 28093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-concat 14202  df-s1 14229  df-s2 14489  df-s3 14490  df-wwlks 28096  df-wwlksn 28097  df-wwlksnon 28098

This theorem is referenced by:  elwwlks2on  28225  frgr2wwlk1  28594