Theorem wlkswwlksf1o 29947

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 6853	. . . . . 6 ⊢ (1st ‘𝑤) ∈ V
2		breq1 5088	. . . . . 6 ⊢ (𝑓 = (1st ‘𝑤) → (𝑓(Walks‘𝐺)(2nd ‘𝑤) ↔ (1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤)))
3	1, 2	spcev 3548	. . . . 5 ⊢ ((1st ‘𝑤)(Walks‘𝐺)(2nd ‘𝑤) → ∃𝑓 𝑓(Walks‘𝐺)(2nd ‘𝑤))
4		wlkiswwlks 29944	. . . . 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 29697	. . . . 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 7066	. 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 6840	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (2nd ‘𝑤) = (2nd ‘𝑥))
13		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ (Walks‘𝐺) → 𝑥 ∈ (Walks‘𝐺))
14		fvexd 6855	. . . . . . . . 9 ⊢ (𝑥 ∈ (Walks‘𝐺) → (2nd ‘𝑥) ∈ V)
15	9, 12, 13, 14	fvmptd3 6971	. . . . . . . 8 ⊢ (𝑥 ∈ (Walks‘𝐺) → (𝐹‘𝑥) = (2nd ‘𝑥))
16		fveq2 6840	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (2nd ‘𝑤) = (2nd ‘𝑦))
17		id 22	. . . . . . . . 9 ⊢ (𝑦 ∈ (Walks‘𝐺) → 𝑦 ∈ (Walks‘𝐺))
18		fvexd 6855	. . . . . . . . 9 ⊢ (𝑦 ∈ (Walks‘𝐺) → (2nd ‘𝑦) ∈ V)
19	9, 16, 17, 18	fvmptd3 6971	. . . . . . . 8 ⊢ (𝑦 ∈ (Walks‘𝐺) → (𝐹‘𝑦) = (2nd ‘𝑦))
20	15, 19	eqeqan12d 2750	. . . . . . 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 29716	. . . . . . . 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 3180	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑥 ∈ (Walks‘𝐺)∀𝑦 ∈ (Walks‘𝐺)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
27		dff13 7209	. . . 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 29944	. . . . . . . . . 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 5086	. . . . . . . . . . 11 ⊢ (𝑓(Walks‘𝐺)𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺))
32		vex 3433	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
33		vex 3433	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
34	32, 33	op2nd 7951	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨𝑓, 𝑦⟩) = 𝑦
35	34	eqcomi 2745	. . . . . . . . . . . 12 ⊢ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)
36		opex 5416	. . . . . . . . . . . . 13 ⊢ ⟨𝑓, 𝑦⟩ ∈ V
37		eleq1 2824	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑥 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑦⟩ ∈ (Walks‘𝐺)))
38		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝑓, 𝑦⟩))
39	38	eqeq2d 2747	. . . . . . . . . . . . . 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 3548	. . . . . . . . . . . 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 3062	. . . . . . 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 2747	. . . . . . 7 ⊢ (𝑥 ∈ (Walks‘𝐺) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = (2nd ‘𝑥)))
50	49	rexbiia 3082	. . . . . 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 3129	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(Walks‘𝐺)⟶(WWalks‘𝐺)) → ∀𝑦 ∈ (WWalks‘𝐺)∃𝑥 ∈ (Walks‘𝐺)𝑦 = (𝐹‘𝑥))
53		dffo3 7054	. . . 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 6505	. . 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 3051  ∃wrex 3061  Vcvv 3429  ⟨cop 4573   class class class wbr 5085   ↦ cmpt 5166  ⟶wf 6494  –1-1→wf1 6495  –onto→wfo 6496  –1-1-onto→wf1o 6497  ‘cfv 6498  1^st c1st 7940  2^nd c2nd 7941  USPGraphcuspgr 29217  Walkscwlks 29665  WWalkscwwlks 29893

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-edg 29117  df-uhgr 29127  df-upgr 29151  df-uspgr 29219  df-wlks 29668  df-wwlks 29898

This theorem is referenced by:  wlkswwlksen  29948  wlknwwlksnbij  29956