Theorem elwwlks2ons3 27455

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 27454	. . . 4 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
4		anass 461	. . . 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 575	. . 3 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉))
6		simpr 477	. . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊‘1) ∈ 𝑉)
7		s3eq2 14088	. . . . . 6 ⊢ (𝑏 = (𝑊‘1) → ⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩)
8		eqeq2 2783	. . . . . . 7 ⊢ (⟨“𝐴𝑏𝐶”⟩ = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ↔ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩))
9		eleq1 2847	. . . . . . 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 621	. . . . . 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 474	. . . 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 477	. . . . . 6 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩) → 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩)
14		eleq1 2847	. . . . . . 7 ⊢ (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐴(𝑊‘1)𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
15	14	biimpac 471	. . . . . 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 504	. . . . 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 473	. . . 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 3536	. . 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 2847	. . . . 5 ⊢ (⟨“𝐴𝑏𝐶”⟩ = 𝑊 → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
21	20	eqcoms 2780	. . . 4 ⊢ (𝑊 = ⟨“𝐴𝑏𝐶”⟩ → (⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
22	21	biimpa 469	. . 3 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
23	22	rexlimivw 3221	. 2 ⊢ (∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))
24	19, 23	impbii 201	1 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))

Syntax hints:   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∃wrex 3083  ‘cfv 6182  (class class class)co 6970  1c1 10330  2c2 11489  ⟨“cs3 14060  Vtxcvtx 26478   WWalksNOn cwwlksnon 27307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10385  ax-resscn 10386  ax-1cn 10387  ax-icn 10388  ax-addcl 10389  ax-addrcl 10390  ax-mulcl 10391  ax-mulrcl 10392  ax-mulcom 10393  ax-addass 10394  ax-mulass 10395  ax-distr 10396  ax-i2m1 10397  ax-1ne0 10398  ax-1rid 10399  ax-rnegex 10400  ax-rrecex 10401  ax-cnre 10402  ax-pre-lttri 10403  ax-pre-lttrn 10404  ax-pre-ltadd 10405  ax-pre-mulgt0 10406

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7495  df-2nd 7496  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-1o 7899  df-oadd 7903  df-er 8083  df-map 8202  df-en 8301  df-dom 8302  df-sdom 8303  df-fin 8304  df-card 9156  df-pnf 10470  df-mnf 10471  df-xr 10472  df-ltxr 10473  df-le 10474  df-sub 10666  df-neg 10667  df-nn 11434  df-2 11497  df-3 11498  df-n0 11702  df-z 11788  df-uz 12053  df-fz 12703  df-fzo 12844  df-hash 13500  df-word 13667  df-concat 13728  df-s1 13753  df-s2 14066  df-s3 14067  df-wwlks 27310  df-wwlksn 27311  df-wwlksnon 27312

This theorem is referenced by:  elwwlks2on  27459  frgr2wwlk1  27857