Theorem wlkswwlksf1o 29970

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 6857	. . . . . 6 ⊢ (1st ‘𝑤) ∈ V
2		breq1 5103	. . . . . 6 ⊢ (𝑓 = (1st ‘𝑤) → (𝑓(Walks‘𝐺)(2nd ‘𝑤) ↔ (1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤)))
3	1, 2	spcev 3562	. . . . 5 ⊢ ((1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤) → ∃𝑓 𝑓(Walks‘𝐺)(2nd ‘𝑤))
4		wlkiswwlks 29967	. . . . 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 29720	. . . . 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 7070	. 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 6844	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (2nd ‘𝑤) = (2nd ‘𝑥))
13		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ (Walks‘𝐺) → 𝑥 ∈ (Walks‘𝐺))
14		fvexd 6859	. . . . . . . . 9 ⊢ (𝑥 ∈ (Walks‘𝐺) → (2nd ‘𝑥) ∈ V)
15	9, 12, 13, 14	fvmptd3 6975	. . . . . . . 8 ⊢ (𝑥 ∈ (Walks‘𝐺) → (𝐹‘𝑥) = (2nd ‘𝑥))
16		fveq2 6844	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (2nd ‘𝑤) = (2nd ‘𝑦))
17		id 22	. . . . . . . . 9 ⊢ (𝑦 ∈ (Walks‘𝐺) → 𝑦 ∈ (Walks‘𝐺))
18		fvexd 6859	. . . . . . . . 9 ⊢ (𝑦 ∈ (Walks‘𝐺) → (2nd ‘𝑦) ∈ V)
19	9, 16, 17, 18	fvmptd3 6975	. . . . . . . 8 ⊢ (𝑦 ∈ (Walks‘𝐺) → (𝐹‘𝑦) = (2nd ‘𝑦))
20	15, 19	eqeqan12d 2751	. . . . . . 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 29739	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → 𝑥 = 𝑦)
23	22	ad4ant134 1176	. . . . . . 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 3181	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑥 ∈ (Walks‘𝐺)∀𝑦 ∈ (Walks‘𝐺)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
27		dff13 7212	. . . 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 584	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–1-1→(WWalks‘𝐺))
29		wlkiswwlks 29967	. . . . . . . . . 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 5101	. . . . . . . . . . 11 ⊢ (𝑓(Walks‘𝐺)𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺))
32		vex 3446	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
33		vex 3446	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
34	32, 33	op2nd 7954	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨𝑓, 𝑦⟩) = 𝑦
35	34	eqcomi 2746	. . . . . . . . . . . 12 ⊢ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)
36		opex 5421	. . . . . . . . . . . . 13 ⊢ ⟨𝑓, 𝑦⟩ ∈ V
37		eleq1 2825	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑥 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺)))
38		fveq2 6844	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝑓, 𝑦⟩))
39	38	eqeq2d 2748	. . . . . . . . . . . . . 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 3562	. . . . . . . . . . . 12 ⊢ ((⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)) → ∃𝑥(𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
42	35, 41	mpan2 692	. . . . . . . . . . 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 1932	. . . . . . . . 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 3063	. . . . . . 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 2748	. . . . . . 7 ⊢ (𝑥 ∈ (Walks‘𝐺) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = (2nd ‘𝑥)))
50	49	rexbiia 3083	. . . . . 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 3130	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥))
53		dffo3 7058	. . . 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 584	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–onto→(WWalks‘𝐺))
55		df-f1o 6509	. . 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 584	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
57	10, 56	mpdan 688	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442  ⟨cop 4588   class class class wbr 5100   ↦ cmpt 5181  ⟶wf 6498  –1-1→wf1 6499  –onto→wfo 6500  –1-1-onto→wf1o 6501  ‘cfv 6502  1^st c1st 7943  2^nd c2nd 7944  USPGraphcuspgr 29239  Walkscwlks 29688  WWalkscwwlks 29916

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-oadd 8413  df-er 8647  df-map 8779  df-pm 8780  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-dju 9827  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-n0 12416  df-xnn0 12489  df-z 12503  df-uz 12766  df-fz 13438  df-fzo 13585  df-hash 14268  df-word 14451  df-edg 29139  df-uhgr 29149  df-upgr 29173  df-uspgr 29241  df-wlks 29691  df-wwlks 29921

This theorem is referenced by:  wlkswwlksen  29971  wlknwwlksnbij  29979