Theorem elwspths2on 30563

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 30459	. . . 4 ⊢ (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑊))
2	1	biimpi 219	. . 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 30562	. . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → 𝑊 = ⟨“𝐴𝑏𝐶”⟩)
6		eleq1 2851	. . . . . . . . . . . . . 14 ⊢ (𝑊 = ⟨“𝐴𝑏𝐶”⟩ → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))
7	6	biimpa 482	. . . . . . . . . . . . 13 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))
8	5, 7	jca 521	. . . . . . . . . . . 12 ⊢ ((𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)) → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))
9	8	ex 418	. . . . . . . . . . 11 ⊢ (𝑊 = ⟨“𝐴𝑏𝐶”⟩ → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
10	9	adantr 486	. . . . . . . . . 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 33	. . . . . . . . 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 3180	. . . . . . . 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 28	. . . . . . 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 96	. . . . . 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 243	. . . . 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 416	. . . 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 87	. . 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 46	. 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 483	. . . 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 3166	. 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 215	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2146  ∃wrex 3089   class class class wbr 5105  ‘cfv 6546  (class class class)co 7427  2c2 12407  ♯chash 14484  ⟨“cs3 15004  Vtxcvtx 29586  UPGraphcupgr 29670  Walkscwlks 30189  SPathsOncspthson 30310   WWalksNOn cwwlksnon 30428   WSPathsNOn cwwspthsnon 30430

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-ac2 10551  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-se 5608  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-2o 8488  df-oadd 8491  df-er 8728  df-map 8860  df-pm 8861  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9992  df-card 10030  df-ac 10205  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-nn 12346  df-2 12415  df-3 12416  df-n0 12617  df-xnn0 12690  df-z 12704  df-uz 12976  df-fz 13650  df-fzo 13799  df-hash 14485  df-word 14669  df-concat 14726  df-s1 14753  df-s2 15010  df-s3 15011  df-edg 29638  df-uhgr 29648  df-upgr 29672  df-wlks 30192  df-wwlks 30431  df-wwlksn 30432  df-wwlksnon 30433  df-wspthsnon 30435

This theorem is used by:  elwspths2spth  30571