Theorem wlknwwlksnbij 27668

Description: The mapping (𝑡 ∈ 𝑇 ↦ (2^nd ‘𝑡)) is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length in a simple pseudograph. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 5-Aug-2022.)

Hypotheses

Ref	Expression
wlknwwlksnbij.t	⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}
wlknwwlksnbij.w	⊢ 𝑊 = (𝑁 WWalksN 𝐺)
wlknwwlksnbij.f	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))

Assertion

Ref	Expression
wlknwwlksnbij	⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1-onto→𝑊)

Distinct variable groups:   𝐺,𝑝,𝑡   𝑁,𝑝,𝑡   𝑡,𝑇

Allowed substitution hints:   𝑇(𝑝)   𝐹(𝑡,𝑝)   𝑊(𝑡,𝑝)

Proof of Theorem wlknwwlksnbij

Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2823	. . 3 ⊢ (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝))
2	1	wlkswwlksf1o 27659	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
3	2	adantr 483	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
4		fveqeq2 6681	. . . . 5 ⊢ (𝑞 = (2nd ‘𝑝) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd ‘𝑝)) = (𝑁 + 1)))
5	4	3ad2ant3 1131	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd ‘𝑝)) = (𝑁 + 1)))
6		wlkcpr 27412	. . . . . . 7 ⊢ (𝑝 ∈ (Walks‘𝐺) ↔ (1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝))
7		wlklenvp1 27402	. . . . . . . 8 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1))
8		eqeq1 2827	. . . . . . . . . 10 ⊢ ((♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ ((♯‘(1st ‘𝑝)) + 1) = (𝑁 + 1)))
9		wlkcl 27399	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(1st ‘𝑝)) ∈ ℕ0)
10	9	nn0cnd 11960	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(1st ‘𝑝)) ∈ ℂ)
11	10	adantr 483	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (♯‘(1st ‘𝑝)) ∈ ℂ)
12		nn0cn 11910	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
13	12	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ)
14	13	adantl 484	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 𝑁 ∈ ℂ)
15		1cnd 10638	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 1 ∈ ℂ)
16	11, 14, 15	addcan2d 10846	. . . . . . . . . 10 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (((♯‘(1st ‘𝑝)) + 1) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
17	8, 16	sylan9bbr 513	. . . . . . . . 9 ⊢ ((((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) ∧ (♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1)) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
18	17	exp31 422	. . . . . . . 8 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))))
19	7, 18	mpid 44	. . . . . . 7 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁)))
20	6, 19	sylbi 219	. . . . . 6 ⊢ (𝑝 ∈ (Walks‘𝐺) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁)))
21	20	impcom 410	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺)) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
22	21	3adant3 1128	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
23	5, 22	bitrd 281	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
24	1, 3, 23	f1oresrab 6891	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}):{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}–1-1-onto→{𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
25		wlknwwlksnbij.f	. . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))
26		wlknwwlksnbij.t	. . . . . . 7 ⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}
27	26	mpteq1i 5158	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ↦ (2nd ‘𝑡))
28		ssrab2 4058	. . . . . . 7 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ⊆ (Walks‘𝐺)
29		resmpt 5907	. . . . . . 7 ⊢ ({𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ⊆ (Walks‘𝐺) → ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ↦ (2nd ‘𝑡)))
30	28, 29	ax-mp 5	. . . . . 6 ⊢ ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ↦ (2nd ‘𝑡))
31		fveq2 6672	. . . . . . . 8 ⊢ (𝑡 = 𝑝 → (2nd ‘𝑡) = (2nd ‘𝑝))
32	31	cbvmptv 5171	. . . . . . 7 ⊢ (𝑡 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑡)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝))
33	32	reseq1i 5851	. . . . . 6 ⊢ ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁})
34	27, 30, 33	3eqtr2i 2852	. . . . 5 ⊢ (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁})
35	34	a1i 11	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}))
36	25, 35	syl5eq 2870	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹 = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}))
37	26	a1i 11	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁})
38		wlknwwlksnbij.w	. . . 4 ⊢ 𝑊 = (𝑁 WWalksN 𝐺)
39		wwlksn 27617	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
40	39	adantl 484	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
41	38, 40	syl5eq 2870	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑊 = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
42	36, 37, 41	f1oeq123d 6612	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝐹:𝑇–1-1-onto→𝑊 ↔ ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}):{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}–1-1-onto→{𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)}))
43	24, 42	mpbird 259	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1-onto→𝑊)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {crab 3144   ⊆ wss 3938   class class class wbr 5068   ↦ cmpt 5148   ↾ cres 5559  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  1^st c1st 7689  2^nd c2nd 7690  ℂcc 10537  1c1 10540   + caddc 10542  ℕ₀cn0 11900  ♯chash 13693  USPGraphcuspgr 26935  Walkscwlks 27380  WWalkscwwlks 27605   WWalksN cwwlksn 27606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-dju 9332  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-fz 12896  df-fzo 13037  df-hash 13694  df-word 13865  df-edg 26835  df-uhgr 26845  df-upgr 26869  df-uspgr 26937  df-wlks 27383  df-wwlks 27610  df-wwlksn 27611

This theorem is referenced by:  wlknwwlksnen  27669  wlksnwwlknvbij  27689