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Theorem fusgreg2wsp 41499
Description: In a finite simple graph, the set of all paths of length 2 is the union of all the paths of length 2 over the vertices which are in the middle of such a path. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 18-May-2021.)
Hypotheses
Ref Expression
frgrhash2wsp.v 𝑉 = (Vtx‘𝐺)
fusgreg2wsp.m 𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
Assertion
Ref Expression
fusgreg2wsp (𝐺 ∈ FinUSGraph → (2 WSPathsN 𝐺) = 𝑥𝑉 (𝑀𝑥))
Distinct variable groups:   𝐺,𝑎   𝑉,𝑎   𝑥,𝐺,𝑤,𝑎   𝑥,𝑉,𝑤,𝑎
Allowed substitution hints:   𝑀(𝑥,𝑤,𝑎)

Proof of Theorem fusgreg2wsp
Dummy variables 𝑝 𝑐 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iswspthn 41046 . . . . . . 7 (𝑝 ∈ (2 WSPathsN 𝐺) ↔ (𝑝 ∈ (2 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑝))
21a1i 11 . . . . . 6 (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) ↔ (𝑝 ∈ (2 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑝)))
3 frgrhash2wsp.v . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
43elwwlks2s3 41168 . . . . . . . 8 (𝑝 ∈ (2 WWalkSN 𝐺) → ∃𝑎𝑉𝑥𝑉𝑐𝑉 𝑝 = ⟨“𝑎𝑥𝑐”⟩)
5 fveq1 6084 . . . . . . . . . . . 12 (𝑝 = ⟨“𝑎𝑥𝑐”⟩ → (𝑝‘1) = (⟨“𝑎𝑥𝑐”⟩‘1))
6 vex 3172 . . . . . . . . . . . . 13 𝑥 ∈ V
7 s3fv1 13430 . . . . . . . . . . . . 13 (𝑥 ∈ V → (⟨“𝑎𝑥𝑐”⟩‘1) = 𝑥)
86, 7ax-mp 5 . . . . . . . . . . . 12 (⟨“𝑎𝑥𝑐”⟩‘1) = 𝑥
95, 8syl6eq 2656 . . . . . . . . . . 11 (𝑝 = ⟨“𝑎𝑥𝑐”⟩ → (𝑝‘1) = 𝑥)
109rexlimivw 3007 . . . . . . . . . 10 (∃𝑐𝑉 𝑝 = ⟨“𝑎𝑥𝑐”⟩ → (𝑝‘1) = 𝑥)
1110reximi 2990 . . . . . . . . 9 (∃𝑥𝑉𝑐𝑉 𝑝 = ⟨“𝑎𝑥𝑐”⟩ → ∃𝑥𝑉 (𝑝‘1) = 𝑥)
1211rexlimivw 3007 . . . . . . . 8 (∃𝑎𝑉𝑥𝑉𝑐𝑉 𝑝 = ⟨“𝑎𝑥𝑐”⟩ → ∃𝑥𝑉 (𝑝‘1) = 𝑥)
134, 12syl 17 . . . . . . 7 (𝑝 ∈ (2 WWalkSN 𝐺) → ∃𝑥𝑉 (𝑝‘1) = 𝑥)
1413adantr 479 . . . . . 6 ((𝑝 ∈ (2 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑝) → ∃𝑥𝑉 (𝑝‘1) = 𝑥)
152, 14syl6bi 241 . . . . 5 (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) → ∃𝑥𝑉 (𝑝‘1) = 𝑥))
1615pm4.71rd 664 . . . 4 (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) ↔ (∃𝑥𝑉 (𝑝‘1) = 𝑥𝑝 ∈ (2 WSPathsN 𝐺))))
17 ancom 464 . . . . . . 7 ((𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ ((𝑝‘1) = 𝑥𝑝 ∈ (2 WSPathsN 𝐺)))
1817rexbii 3019 . . . . . 6 (∃𝑥𝑉 (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ ∃𝑥𝑉 ((𝑝‘1) = 𝑥𝑝 ∈ (2 WSPathsN 𝐺)))
19 r19.41v 3066 . . . . . 6 (∃𝑥𝑉 ((𝑝‘1) = 𝑥𝑝 ∈ (2 WSPathsN 𝐺)) ↔ (∃𝑥𝑉 (𝑝‘1) = 𝑥𝑝 ∈ (2 WSPathsN 𝐺)))
2018, 19bitr2i 263 . . . . 5 ((∃𝑥𝑉 (𝑝‘1) = 𝑥𝑝 ∈ (2 WSPathsN 𝐺)) ↔ ∃𝑥𝑉 (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥))
2120a1i 11 . . . 4 (𝐺 ∈ FinUSGraph → ((∃𝑥𝑉 (𝑝‘1) = 𝑥𝑝 ∈ (2 WSPathsN 𝐺)) ↔ ∃𝑥𝑉 (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥)))
22 simpr 475 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑥𝑉) → 𝑥𝑉)
23 ovex 6552 . . . . . . . . 9 (2 WSPathsN 𝐺) ∈ V
2423rabex 4732 . . . . . . . 8 {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥} ∈ V
25 eqeq2 2617 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝑤‘1) = 𝑎 ↔ (𝑤‘1) = 𝑥))
2625rabbidv 3160 . . . . . . . . 9 (𝑎 = 𝑥 → {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎} = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥})
27 fusgreg2wsp.m . . . . . . . . 9 𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
2826, 27fvmptg 6171 . . . . . . . 8 ((𝑥𝑉 ∧ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥} ∈ V) → (𝑀𝑥) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥})
2922, 24, 28sylancl 692 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑥𝑉) → (𝑀𝑥) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥})
3029eleq2d 2669 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑥𝑉) → (𝑝 ∈ (𝑀𝑥) ↔ 𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥}))
31 fveq1 6084 . . . . . . . 8 (𝑤 = 𝑝 → (𝑤‘1) = (𝑝‘1))
3231eqeq1d 2608 . . . . . . 7 (𝑤 = 𝑝 → ((𝑤‘1) = 𝑥 ↔ (𝑝‘1) = 𝑥))
3332elrab 3327 . . . . . 6 (𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥} ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥))
3430, 33syl6rbb 275 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑥𝑉) → ((𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ 𝑝 ∈ (𝑀𝑥)))
3534rexbidva 3027 . . . 4 (𝐺 ∈ FinUSGraph → (∃𝑥𝑉 (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ ∃𝑥𝑉 𝑝 ∈ (𝑀𝑥)))
3616, 21, 353bitrd 292 . . 3 (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥𝑉 𝑝 ∈ (𝑀𝑥)))
37 eliun 4451 . . 3 (𝑝 𝑥𝑉 (𝑀𝑥) ↔ ∃𝑥𝑉 𝑝 ∈ (𝑀𝑥))
3836, 37syl6bbr 276 . 2 (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) ↔ 𝑝 𝑥𝑉 (𝑀𝑥)))
3938eqrdv 2604 1 (𝐺 ∈ FinUSGraph → (2 WSPathsN 𝐺) = 𝑥𝑉 (𝑀𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1976  wrex 2893  {crab 2896  Vcvv 3169   ciun 4446   class class class wbr 4574  cmpt 4634  cfv 5787  (class class class)co 6524  1c1 9790  2c2 10914  ⟨“cs3 13381  Vtxcvtx 40228   FinUSGraph cfusgr 40534  SPathScspths 40919   WWalkSN cwwlksn 41028   WSPathsN cwwspthsn 41030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-map 7720  df-pm 7721  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-card 8622  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-2 10923  df-3 10924  df-n0 11137  df-z 11208  df-uz 11517  df-fz 12150  df-fzo 12287  df-hash 12932  df-word 13097  df-concat 13099  df-s1 13100  df-s2 13387  df-s3 13388  df-wwlks 41032  df-wwlksn 41033  df-wspthsn 41035
This theorem is referenced by:  fusgreghash2wsp  41501
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