Theorem wlkswwlksf1o 29828

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 6899	. . . . . 6 ⊢ (1st ‘𝑤) ∈ V
2		breq1 5126	. . . . . 6 ⊢ (𝑓 = (1st ‘𝑤) → (𝑓(Walks‘𝐺)(2nd ‘𝑤) ↔ (1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤)))
3	1, 2	spcev 3589	. . . . 5 ⊢ ((1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤) → ∃𝑓 𝑓(Walks‘𝐺)(2nd ‘𝑤))
4		wlkiswwlks 29825	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)(2nd ‘𝑤) ↔ (2nd ‘𝑤) ∈ (WWalks‘𝐺)))
5	3, 4	imbitrid 244	. . . 4 ⊢ (𝐺 ∈ USPGraph → ((1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤) → (2nd ‘𝑤) ∈ (WWalks‘𝐺)))
6		wlkcpr 29576	. . . . 5 ⊢ (𝑤 ∈ (Walks‘𝐺) ↔ (1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤))
7	6	biimpi 216	. . . 4 ⊢ (𝑤 ∈ (Walks‘𝐺) → (1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤))
8	5, 7	impel 505	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (Walks‘𝐺)) → (2nd ‘𝑤) ∈ (WWalks‘𝐺))
9		wlkswwlksf1o.f	. . 3 ⊢ 𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑤))
10	8, 9	fmptd 7114	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺))
11		simpr 484	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺))
12		fveq2 6886	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (2nd ‘𝑤) = (2nd ‘𝑥))
13		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ (Walks‘𝐺) → 𝑥 ∈ (Walks‘𝐺))
14		fvexd 6901	. . . . . . . . 9 ⊢ (𝑥 ∈ (Walks‘𝐺) → (2nd ‘𝑥) ∈ V)
15	9, 12, 13, 14	fvmptd3 7019	. . . . . . . 8 ⊢ (𝑥 ∈ (Walks‘𝐺) → (𝐹‘𝑥) = (2nd ‘𝑥))
16		fveq2 6886	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (2nd ‘𝑤) = (2nd ‘𝑦))
17		id 22	. . . . . . . . 9 ⊢ (𝑦 ∈ (Walks‘𝐺) → 𝑦 ∈ (Walks‘𝐺))
18		fvexd 6901	. . . . . . . . 9 ⊢ (𝑦 ∈ (Walks‘𝐺) → (2nd ‘𝑦) ∈ V)
19	9, 16, 17, 18	fvmptd3 7019	. . . . . . . 8 ⊢ (𝑦 ∈ (Walks‘𝐺) → (𝐹‘𝑦) = (2nd ‘𝑦))
20	15, 19	eqeqan12d 2748	. . . . . . 7 ⊢ ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (2nd ‘𝑥) = (2nd ‘𝑦)))
21	20	adantl 481	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (2nd ‘𝑥) = (2nd ‘𝑦)))
22		uspgr2wlkeqi 29595	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → 𝑥 = 𝑦)
23	22	ad4ant134 1174	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → 𝑥 = 𝑦)
24	23	ex 412	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((2nd ‘𝑥) = (2nd ‘𝑦) → 𝑥 = 𝑦))
25	21, 24	sylbid 240	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
26	25	ralrimivva 3189	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑥 ∈ (Walks‘𝐺)∀𝑦 ∈ (Walks‘𝐺)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
27		dff13 7257	. . . 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 583	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–1-1→(WWalks‘𝐺))
29		wlkiswwlks 29825	. . . . . . . . . 10 ⊢ (𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)𝑦 ↔ 𝑦 ∈ (WWalks‘𝐺)))
30	29	adantr 480	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → (∃𝑓 𝑓(Walks‘𝐺)𝑦 ↔ 𝑦 ∈ (WWalks‘𝐺)))
31		df-br 5124	. . . . . . . . . . 11 ⊢ (𝑓(Walks‘𝐺)𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺))
32		vex 3467	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
33		vex 3467	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
34	32, 33	op2nd 8005	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨𝑓, 𝑦⟩) = 𝑦
35	34	eqcomi 2743	. . . . . . . . . . . 12 ⊢ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)
36		opex 5449	. . . . . . . . . . . . 13 ⊢ ⟨𝑓, 𝑦⟩ ∈ V
37		eleq1 2821	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑥 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺)))
38		fveq2 6886	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝑓, 𝑦⟩))
39	38	eqeq2d 2745	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑦 = (2nd ‘𝑥) ↔ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)))
40	37, 39	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → ((𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)) ↔ (⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩))))
41	36, 40	spcev 3589	. . . . . . . . . . . 12 ⊢ ((⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
42	35, 41	mpan2 691	. . . . . . . . . . 11 ⊢ (⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
43	31, 42	sylbi 217	. . . . . . . . . 10 ⊢ (𝑓(Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
44	43	exlimiv 1929	. . . . . . . . 9 ⊢ (∃𝑓 𝑓(Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
45	30, 44	biimtrrdi 254	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → (𝑦 ∈ (WWalks‘𝐺) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥))))
46	45	imp 406	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
47		df-rex 3060	. . . . . . 7 ⊢ (∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd ‘𝑥) ↔ ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
48	46, 47	sylibr 234	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd ‘𝑥))
49	15	eqeq2d 2745	. . . . . . 7 ⊢ (𝑥 ∈ (Walks‘𝐺) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = (2nd ‘𝑥)))
50	49	rexbiia 3080	. . . . . 6 ⊢ (∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (2nd ‘𝑥))
51	48, 50	sylibr 234	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) ∧ 𝑦 ∈ (WWalks‘𝐺)) → ∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥))
52	51	ralrimiva 3133	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥))
53		dffo3 7102	. . . 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 583	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺))
55		df-f1o 6548	. . 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 583	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
57	10, 56	mpdan 687	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059  Vcvv 3463  ⟨cop 4612   class class class wbr 5123   ↦ cmpt 5205  ⟶wf 6537  –1-1→wf1 6538  –onto→wfo 6539  –1-1-onto→wf1o 6540  ‘cfv 6541  1^st c1st 7994  2^nd c2nd 7995  USPGraphcuspgr 29094  Walkscwlks 29543  WWalkscwwlks 29774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-oadd 8492  df-er 8727  df-map 8850  df-pm 8851  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-dju 9923  df-card 9961  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-2 12311  df-n0 12510  df-xnn0 12583  df-z 12597  df-uz 12861  df-fz 13530  df-fzo 13677  df-hash 14353  df-word 14536  df-edg 28994  df-uhgr 29004  df-upgr 29028  df-uspgr 29096  df-wlks 29546  df-wwlks 29779

This theorem is referenced by:  wlkswwlksen  29829  wlknwwlksnbij  29837