Theorem wlknwwlksnbij 30410

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 2760	. . 3 ⊢ (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝))
2	1	wlkswwlksf1o 30401	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
3	2	adantr 486	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
4		fveqeq2 6882	. . . . 5 ⊢ (𝑞 = (2nd ‘𝑝) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd ‘𝑝)) = (𝑁 + 1)))
5	4	3ad2ant3 1153	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd ‘𝑝)) = (𝑁 + 1)))
6		wlkcpr 30142	. . . . . . 7 ⊢ (𝑝 ∈ (Walks‘𝐺) ↔ (1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝))
7		wlklenvp1 30132	. . . . . . . 8 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1))
8		eqeq1 2764	. . . . . . . . . 10 ⊢ ((♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ ((♯‘(1st ‘𝑝)) + 1) = (𝑁 + 1)))
9		wlkcl 30129	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(1st ‘𝑝)) ∈ ℕ0)
10	9	nn0cnd 12638	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(1st ‘𝑝)) ∈ ℂ)
11	10	adantr 486	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (♯‘(1st ‘𝑝)) ∈ ℂ)
12		nn0cn 12585	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
13	12	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ)
14	13	adantl 487	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 𝑁 ∈ ℂ)
15		1cnd 11273	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 1 ∈ ℂ)
16	11, 14, 15	addcan2d 11485	. . . . . . . . . 10 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (((♯‘(1st ‘𝑝)) + 1) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
17	8, 16	sylan9bbr 520	. . . . . . . . 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 425	. . . . . . . 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 45	. . . . . . 7 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁)))
20	6, 19	sylbi 220	. . . . . 6 ⊢ (𝑝 ∈ (Walks‘𝐺) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁)))
21	20	impcom 413	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺)) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
22	21	3adant3 1150	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
23	5, 22	bitrd 282	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
24	1, 3, 23	f1oresrab 7116	. 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 5195	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ↦ (2nd ‘𝑡))
28		ssrab2 4027	. . . . . . 7 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ⊆ (Walks‘𝐺)
29		resmpt 6027	. . . . . . 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 6873	. . . . . . . 8 ⊢ (𝑡 = 𝑝 → (2nd ‘𝑡) = (2nd ‘𝑝))
32	31	cbvmptv 5208	. . . . . . 7 ⊢ (𝑡 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑡)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝))
33	32	reseq1i 5962	. . . . . 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 2789	. . . . 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 2807	. . 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 30359	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
40	39	adantl 487	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
41	38, 40	eqtrid 2807	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑊 = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
42	36, 37, 41	f1oeq123d 6806	. 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 260	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1-onto→𝑊)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3412   ⊆ wss 3898   class class class wbr 5102   ↦ cmpt 5185   ↾ cres 5649  –1-1-onto→wf1o 6526  ‘cfv 6527  (class class class)co 7408  1^st c1st 7982  2^nd c2nd 7983  ℂcc 11169  1c1 11172   + caddc 11174  ℕ₀cn0 12575  ♯chash 14441  USPGraphcuspgr 29662  Walkscwlks 30110  WWalkscwwlks 30347   WWalksN cwwlksn 30348

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-oadd 8458  df-er 8695  df-map 8827  df-pm 8828  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-dju 9953  df-card 9991  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-n0 12576  df-xnn0 12649  df-z 12663  df-uz 12935  df-fz 13609  df-fzo 13757  df-hash 14442  df-word 14626  df-edg 29559  df-uhgr 29569  df-upgr 29593  df-uspgr 29664  df-wlks 30113  df-wwlks 30352  df-wwlksn 30353

This theorem is used by:  wlknwwlksnen  30411  wlksnwwlknvbij  30430