Theorem wlknwwlksnbij 28154

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 2738	. . 3 ⊢ (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝))
2	1	wlkswwlksf1o 28145	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
3	2	adantr 480	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
4		fveqeq2 6765	. . . . 5 ⊢ (𝑞 = (2nd ‘𝑝) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd ‘𝑝)) = (𝑁 + 1)))
5	4	3ad2ant3 1133	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd ‘𝑝)) = (𝑁 + 1)))
6		wlkcpr 27898	. . . . . . 7 ⊢ (𝑝 ∈ (Walks‘𝐺) ↔ (1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝))
7		wlklenvp1 27888	. . . . . . . 8 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1))
8		eqeq1 2742	. . . . . . . . . 10 ⊢ ((♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ ((♯‘(1st ‘𝑝)) + 1) = (𝑁 + 1)))
9		wlkcl 27885	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(1st ‘𝑝)) ∈ ℕ0)
10	9	nn0cnd 12225	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(1st ‘𝑝)) ∈ ℂ)
11	10	adantr 480	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (♯‘(1st ‘𝑝)) ∈ ℂ)
12		nn0cn 12173	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
13	12	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ)
14	13	adantl 481	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 𝑁 ∈ ℂ)
15		1cnd 10901	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 1 ∈ ℂ)
16	11, 14, 15	addcan2d 11109	. . . . . . . . . 10 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (((♯‘(1st ‘𝑝)) + 1) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
17	8, 16	sylan9bbr 510	. . . . . . . . 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 419	. . . . . . . 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 216	. . . . . 6 ⊢ (𝑝 ∈ (Walks‘𝐺) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁)))
21	20	impcom 407	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺)) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
22	21	3adant3 1130	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
23	5, 22	bitrd 278	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
24	1, 3, 23	f1oresrab 6981	. 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 5166	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ↦ (2nd ‘𝑡))
28		ssrab2 4009	. . . . . . 7 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ⊆ (Walks‘𝐺)
29		resmpt 5934	. . . . . . 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 6756	. . . . . . . 8 ⊢ (𝑡 = 𝑝 → (2nd ‘𝑡) = (2nd ‘𝑝))
32	31	cbvmptv 5183	. . . . . . 7 ⊢ (𝑡 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑡)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝))
33	32	reseq1i 5876	. . . . . 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 2772	. . . . 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 2791	. . 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 28103	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
40	39	adantl 481	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
41	38, 40	syl5eq 2791	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑊 = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
42	36, 37, 41	f1oeq123d 6694	. 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 256	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1-onto→𝑊)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {crab 3067   ⊆ wss 3883   class class class wbr 5070   ↦ cmpt 5153   ↾ cres 5582  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  ℂcc 10800  1c1 10803   + caddc 10805  ℕ₀cn0 12163  ♯chash 13972  USPGraphcuspgr 27421  Walkscwlks 27866  WWalkscwwlks 28091   WWalksN cwwlksn 28092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-edg 27321  df-uhgr 27331  df-upgr 27355  df-uspgr 27423  df-wlks 27869  df-wwlks 28096  df-wwlksn 28097

This theorem is referenced by:  wlknwwlksnen  28155  wlksnwwlknvbij  28174