Theorem fusgreg2wsp 41500

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 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})

Assertion

Ref	Expression
fusgreg2wsp	⊢ (𝐺 ∈ FinUSGraph → (2 WSPathsN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥))

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.

Step	Hyp	Ref	Expression
1		iswspthn 41047	. . . . . . 7 ⊢ (𝑝 ∈ (2 WSPathsN 𝐺) ↔ (𝑝 ∈ (2 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑝))
2	1	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) ↔ (𝑝 ∈ (2 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑝)))
3		frgrhash2wsp.v	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	elwwlks2s3 41169	. . . . . . . 8 ⊢ (𝑝 ∈ (2 WWalkSN 𝐺) → ∃𝑎 ∈ 𝑉 ∃𝑥 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑝 = ⟨“𝑎𝑥𝑐”⟩)
5		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨“𝑎𝑥𝑐”⟩ → (𝑝‘1) = (⟨“𝑎𝑥𝑐”⟩‘1))
6		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
7		s3fv1 13487	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ V → (⟨“𝑎𝑥𝑐”⟩‘1) = 𝑥)
8	6, 7	ax-mp 5	. . . . . . . . . . . 12 ⊢ (⟨“𝑎𝑥𝑐”⟩‘1) = 𝑥
9	5, 8	syl6eq 2660	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨“𝑎𝑥𝑐”⟩ → (𝑝‘1) = 𝑥)
10	9	rexlimivw 3011	. . . . . . . . . 10 ⊢ (∃𝑐 ∈ 𝑉 𝑝 = ⟨“𝑎𝑥𝑐”⟩ → (𝑝‘1) = 𝑥)
11	10	reximi 2994	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑝 = ⟨“𝑎𝑥𝑐”⟩ → ∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥)
12	11	rexlimivw 3011	. . . . . . . 8 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑥 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑝 = ⟨“𝑎𝑥𝑐”⟩ → ∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥)
13	4, 12	syl 17	. . . . . . 7 ⊢ (𝑝 ∈ (2 WWalkSN 𝐺) → ∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥)
14	13	adantr 480	. . . . . 6 ⊢ ((𝑝 ∈ (2 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑝) → ∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥)
15	2, 14	syl6bi 242	. . . . 5 ⊢ (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) → ∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥))
16	15	pm4.71rd 665	. . . 4 ⊢ (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) ↔ (∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺))))
17		ancom 465	. . . . . . 7 ⊢ ((𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ ((𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺)))
18	17	rexbii 3023	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑉 (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ ∃𝑥 ∈ 𝑉 ((𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺)))
19		r19.41v 3070	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑉 ((𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺)) ↔ (∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺)))
20	18, 19	bitr2i 264	. . . . 5 ⊢ ((∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺)) ↔ ∃𝑥 ∈ 𝑉 (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥))
21	20	a1i 11	. . . 4 ⊢ (𝐺 ∈ FinUSGraph → ((∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺)) ↔ ∃𝑥 ∈ 𝑉 (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥)))
22		simpr 476	. . . . . . . 8 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
23		ovex 6577	. . . . . . . . 9 ⊢ (2 WSPathsN 𝐺) ∈ V
24	23	rabex 4740	. . . . . . . 8 ⊢ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥} ∈ V
25		eqeq2 2621	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((𝑤‘1) = 𝑎 ↔ (𝑤‘1) = 𝑥))
26	25	rabbidv 3164	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎} = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥})
27		fusgreg2wsp.m	. . . . . . . . 9 ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
28	26, 27	fvmptg 6189	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥} ∈ V) → (𝑀‘𝑥) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥})
29	22, 24, 28	sylancl 693	. . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑥 ∈ 𝑉) → (𝑀‘𝑥) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥})
30	29	eleq2d 2673	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑥 ∈ 𝑉) → (𝑝 ∈ (𝑀‘𝑥) ↔ 𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥}))
31		fveq1 6102	. . . . . . . 8 ⊢ (𝑤 = 𝑝 → (𝑤‘1) = (𝑝‘1))
32	31	eqeq1d 2612	. . . . . . 7 ⊢ (𝑤 = 𝑝 → ((𝑤‘1) = 𝑥 ↔ (𝑝‘1) = 𝑥))
33	32	elrab 3331	. . . . . 6 ⊢ (𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥} ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥))
34	30, 33	syl6rbb 276	. . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑥 ∈ 𝑉) → ((𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ 𝑝 ∈ (𝑀‘𝑥)))
35	34	rexbidva 3031	. . . 4 ⊢ (𝐺 ∈ FinUSGraph → (∃𝑥 ∈ 𝑉 (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ ∃𝑥 ∈ 𝑉 𝑝 ∈ (𝑀‘𝑥)))
36	16, 21, 35	3bitrd 293	. . 3 ⊢ (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥 ∈ 𝑉 𝑝 ∈ (𝑀‘𝑥)))
37		eliun 4460	. . 3 ⊢ (𝑝 ∈ ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥) ↔ ∃𝑥 ∈ 𝑉 𝑝 ∈ (𝑀‘𝑥))
38	36, 37	syl6bbr 277	. 2 ⊢ (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) ↔ 𝑝 ∈ ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥)))
39	38	eqrdv 2608	1 ⊢ (𝐺 ∈ FinUSGraph → (2 WSPathsN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  1c1 9816  2c2 10947  ⟨“cs3 13438  Vtxcvtx 25673   FinUSGraph cfusgr 40535  SPathScspths 40920   WWalkSN cwwlksn 41029   WSPathsN cwwspthsn 41031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-wwlks 41033  df-wwlksn 41034  df-wspthsn 41036

This theorem is referenced by:  fusgreghash2wsp  41502