Theorem wlkswwlksf1o 27764

Description: The mapping of (ordinary) walks to their sequences of vertices is a bijection in a simple pseudograph. (Contributed by AV, 6-May-2021.)

Hypothesis

Ref	Expression
wlkswwlksf1o.f	⊢ 𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑤))

Assertion

Ref	Expression
wlkswwlksf1o	⊢ (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))

Distinct variable group:   𝑤,𝐺

Allowed substitution hint:   𝐹(𝑤)

Proof of Theorem wlkswwlksf1o

Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6671	. . . . . 6 ⊢ (1st ‘𝑤) ∈ V
2		breq1 5035	. . . . . 6 ⊢ (𝑓 = (1st ‘𝑤) → (𝑓(Walks‘𝐺)(2nd ‘𝑤) ↔ (1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤)))
3	1, 2	spcev 3525	. . . . 5 ⊢ ((1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤) → ∃𝑓 𝑓(Walks‘𝐺)(2nd ‘𝑤))
4		wlkiswwlks 27761	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)(2nd ‘𝑤) ↔ (2nd ‘𝑤) ∈ (WWalks‘𝐺)))
5	3, 4	syl5ib 247	. . . 4 ⊢ (𝐺 ∈ USPGraph → ((1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤) → (2nd ‘𝑤) ∈ (WWalks‘𝐺)))
6		wlkcpr 27517	. . . . 5 ⊢ (𝑤 ∈ (Walks‘𝐺) ↔ (1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤))
7	6	biimpi 219	. . . 4 ⊢ (𝑤 ∈ (Walks‘𝐺) → (1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤))
8	5, 7	impel 509	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (Walks‘𝐺)) → (2nd ‘𝑤) ∈ (WWalks‘𝐺))
9		wlkswwlksf1o.f	. . 3 ⊢ 𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑤))
10	8, 9	fmptd 6869	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺))
11		simpr 488	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺))
12		fveq2 6658	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (2nd ‘𝑤) = (2nd ‘𝑥))
13		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ (Walks‘𝐺) → 𝑥 ∈ (Walks‘𝐺))
14		fvexd 6673	. . . . . . . . 9 ⊢ (𝑥 ∈ (Walks‘𝐺) → (2nd ‘𝑥) ∈ V)
15	9, 12, 13, 14	fvmptd3 6782	. . . . . . . 8 ⊢ (𝑥 ∈ (Walks‘𝐺) → (𝐹‘𝑥) = (2nd ‘𝑥))
16		fveq2 6658	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (2nd ‘𝑤) = (2nd ‘𝑦))
17		id 22	. . . . . . . . 9 ⊢ (𝑦 ∈ (Walks‘𝐺) → 𝑦 ∈ (Walks‘𝐺))
18		fvexd 6673	. . . . . . . . 9 ⊢ (𝑦 ∈ (Walks‘𝐺) → (2nd ‘𝑦) ∈ V)
19	9, 16, 17, 18	fvmptd3 6782	. . . . . . . 8 ⊢ (𝑦 ∈ (Walks‘𝐺) → (𝐹‘𝑦) = (2nd ‘𝑦))
20	15, 19	eqeqan12d 2775	. . . . . . 7 ⊢ ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (2nd ‘𝑥) = (2nd ‘𝑦)))
21	20	adantl 485	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (2nd ‘𝑥) = (2nd ‘𝑦)))
22		uspgr2wlkeqi 27536	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → 𝑥 = 𝑦)
23	22	ad4ant134 1171	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → 𝑥 = 𝑦)
24	23	ex 416	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((2nd ‘𝑥) = (2nd ‘𝑦) → 𝑥 = 𝑦))
25	21, 24	sylbid 243	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
26	25	ralrimivva 3120	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑥 ∈ (Walks‘𝐺)∀𝑦 ∈ (Walks‘𝐺)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
27		dff13 7005	. . . 4 ⊢ (𝐹:(Walks‘𝐺)–1-1→(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺) ∧ ∀𝑥 ∈ (Walks‘𝐺)∀𝑦 ∈ (Walks‘𝐺)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
28	11, 26, 27	sylanbrc 586	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–1-1→(WWalks‘𝐺))
29		wlkiswwlks 27761	. . . . . . . . . 10 ⊢ (𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)𝑦 ↔ 𝑦 ∈ (WWalks‘𝐺)))
30	29	adantr 484	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → (∃𝑓 𝑓(Walks‘𝐺)𝑦 ↔ 𝑦 ∈ (WWalks‘𝐺)))
31		df-br 5033	. . . . . . . . . . 11 ⊢ (𝑓(Walks‘𝐺)𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺))
32		vex 3413	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
33		vex 3413	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
34	32, 33	op2nd 7702	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨𝑓, 𝑦⟩) = 𝑦
35	34	eqcomi 2767	. . . . . . . . . . . 12 ⊢ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)
36		opex 5324	. . . . . . . . . . . . 13 ⊢ ⟨𝑓, 𝑦⟩ ∈ V
37		eleq1 2839	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑥 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺)))
38		fveq2 6658	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝑓, 𝑦⟩))
39	38	eqeq2d 2769	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑦 = (2nd ‘𝑥) ↔ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)))
40	37, 39	anbi12d 633	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)) ↔ (⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩))))
41	36, 40	spcev 3525	. . . . . . . . . . . 12 ⊢ ((⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
42	35, 41	mpan2 690	. . . . . . . . . . 11 ⊢ (⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
43	31, 42	sylbi 220	. . . . . . . . . 10 ⊢ (𝑓(Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
44	43	exlimiv 1931	. . . . . . . . 9 ⊢ (∃𝑓 𝑓(Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
45	30, 44	syl6bir 257	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → (𝑦 ∈ (WWalks‘𝐺) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥))))
46	45	imp 410	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
47		df-rex 3076	. . . . . . 7 ⊢ (∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd ‘𝑥) ↔ ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
48	46, 47	sylibr 237	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd ‘𝑥))
49	15	eqeq2d 2769	. . . . . . 7 ⊢ (𝑥 ∈ (Walks‘𝐺) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = (2nd ‘𝑥)))
50	49	rexbiia 3174	. . . . . 6 ⊢ (∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd ‘𝑥))
51	48, 50	sylibr 237	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥))
52	51	ralrimiva 3113	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥))
53		dffo3 6859	. . . 4 ⊢ (𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺) ∧ ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥)))
54	11, 52, 53	sylanbrc 586	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺))
55		df-f1o 6342	. . 3 ⊢ (𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺) ↔ (𝐹:(Walks‘𝐺)–1-1→(WWalks‘𝐺) ∧ 𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺)))
56	28, 54, 55	sylanbrc 586	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
57	10, 56	mpdan 686	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  Vcvv 3409  ⟨cop 4528   class class class wbr 5032   ↦ cmpt 5112  ⟶wf 6331  –1-1→wf1 6332  –onto→wfo 6333  –1-1-onto→wf1o 6334  ‘cfv 6335  1^st c1st 7691  2^nd c2nd 7692  USPGraphcuspgr 27040  Walkscwlks 27485  WWalkscwwlks 27710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-2o 8113  df-oadd 8116  df-er 8299  df-map 8418  df-pm 8419  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-dju 9363  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-n0 11935  df-xnn0 12007  df-z 12021  df-uz 12283  df-fz 12940  df-fzo 13083  df-hash 13741  df-word 13914  df-edg 26940  df-uhgr 26950  df-upgr 26974  df-uspgr 27042  df-wlks 27488  df-wwlks 27715

This theorem is referenced by:  wlkswwlksen  27765  wlknwwlksnbij  27773