Theorem wlknwwlksnbij 29944

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 2735	. . 3 ⊢ (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝))
2	1	wlkswwlksf1o 29935	. . . 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 6838	. . . . 5 ⊢ (𝑞 = (2nd ‘𝑝) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd ‘𝑝)) = (𝑁 + 1)))
5	4	3ad2ant3 1136	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd ‘𝑝)) = (𝑁 + 1)))
6		wlkcpr 29685	. . . . . . 7 ⊢ (𝑝 ∈ (Walks‘𝐺) ↔ (1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝))
7		wlklenvp1 29675	. . . . . . . 8 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1))
8		eqeq1 2739	. . . . . . . . . 10 ⊢ ((♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ ((♯‘(1st ‘𝑝)) + 1) = (𝑁 + 1)))
9		wlkcl 29672	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(1st ‘𝑝)) ∈ ℕ0)
10	9	nn0cnd 12489	. . . . . . . . . . . 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 12436	. . . . . . . . . . . . 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 11128	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 1 ∈ ℂ)
16	11, 14, 15	addcan2d 11339	. . . . . . . . . 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 217	. . . . . 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 1133	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
23	5, 22	bitrd 279	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
24	1, 3, 23	f1oresrab 7069	. 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 5165	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ↦ (2nd ‘𝑡))
28		ssrab2 4013	. . . . . . 7 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ⊆ (Walks‘𝐺)
29		resmpt 5991	. . . . . . 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 6829	. . . . . . . 8 ⊢ (𝑡 = 𝑝 → (2nd ‘𝑡) = (2nd ‘𝑝))
32	31	cbvmptv 5178	. . . . . . 7 ⊢ (𝑡 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑡)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝))
33	32	reseq1i 5929	. . . . . 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 2764	. . . . 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	eqtrid 2782	. . 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 29893	. . . . 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	eqtrid 2782	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑊 = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
42	36, 37, 41	f1oeq123d 6763	. 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 257	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1-onto→𝑊)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {crab 3387   ⊆ wss 3885   class class class wbr 5074   ↦ cmpt 5155   ↾ cres 5622  –1-1-onto→wf1o 6486  ‘cfv 6487  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  ℂcc 11025  1c1 11028   + caddc 11030  ℕ₀cn0 12426  ♯chash 14281  USPGraphcuspgr 29205  Walkscwlks 29653  WWalkscwwlks 29881   WWalksN cwwlksn 29882

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 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-oadd 8398  df-er 8632  df-map 8764  df-pm 8765  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-dju 9814  df-card 9852  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-2 12233  df-n0 12427  df-xnn0 12500  df-z 12514  df-uz 12778  df-fz 13451  df-fzo 13598  df-hash 14282  df-word 14465  df-edg 29105  df-uhgr 29115  df-upgr 29139  df-uspgr 29207  df-wlks 29656  df-wwlks 29886  df-wwlksn 29887

This theorem is referenced by:  wlknwwlksnen  29945  wlksnwwlknvbij  29964