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Theorem wlknwwlksnbij 29908
Description: The mapping (𝑡𝑇 ↦ (2nd𝑡)) 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) ∣ (♯‘(1st𝑝)) = 𝑁}
wlknwwlksnbij.w 𝑊 = (𝑁 WWalksN 𝐺)
wlknwwlksnbij.f 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
Assertion
Ref Expression
wlknwwlksnbij ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)
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.
StepHypRef Expression
1 eqid 2737 . . 3 (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝))
21wlkswwlksf1o 29899 . . . 4 (𝐺 ∈ USPGraph → (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
32adantr 480 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
4 fveqeq2 6915 . . . . 5 (𝑞 = (2nd𝑝) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd𝑝)) = (𝑁 + 1)))
543ad2ant3 1136 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd𝑝)) = (𝑁 + 1)))
6 wlkcpr 29647 . . . . . . 7 (𝑝 ∈ (Walks‘𝐺) ↔ (1st𝑝)(Walks‘𝐺)(2nd𝑝))
7 wlklenvp1 29636 . . . . . . . 8 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → (♯‘(2nd𝑝)) = ((♯‘(1st𝑝)) + 1))
8 eqeq1 2741 . . . . . . . . . 10 ((♯‘(2nd𝑝)) = ((♯‘(1st𝑝)) + 1) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ ((♯‘(1st𝑝)) + 1) = (𝑁 + 1)))
9 wlkcl 29633 . . . . . . . . . . . . 13 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → (♯‘(1st𝑝)) ∈ ℕ0)
109nn0cnd 12589 . . . . . . . . . . . 12 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → (♯‘(1st𝑝)) ∈ ℂ)
1110adantr 480 . . . . . . . . . . 11 (((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (♯‘(1st𝑝)) ∈ ℂ)
12 nn0cn 12536 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
1312adantl 481 . . . . . . . . . . . 12 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ)
1413adantl 481 . . . . . . . . . . 11 (((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 𝑁 ∈ ℂ)
15 1cnd 11256 . . . . . . . . . . 11 (((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 1 ∈ ℂ)
1611, 14, 15addcan2d 11465 . . . . . . . . . 10 (((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (((♯‘(1st𝑝)) + 1) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁))
178, 16sylan9bbr 510 . . . . . . . . 9 ((((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) ∧ (♯‘(2nd𝑝)) = ((♯‘(1st𝑝)) + 1)) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁))
1817exp31 419 . . . . . . . 8 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd𝑝)) = ((♯‘(1st𝑝)) + 1) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁))))
197, 18mpid 44 . . . . . . 7 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁)))
206, 19sylbi 217 . . . . . 6 (𝑝 ∈ (Walks‘𝐺) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁)))
2120impcom 407 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺)) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁))
22213adant3 1133 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd𝑝)) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁))
235, 22bitrd 279 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁))
241, 3, 23f1oresrab 7147 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}):{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}–1-1-onto→{𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
25 wlknwwlksnbij.f . . . 4 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
26 wlknwwlksnbij.t . . . . . . 7 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}
2726mpteq1i 5238 . . . . . 6 (𝑡𝑇 ↦ (2nd𝑡)) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ↦ (2nd𝑡))
28 ssrab2 4080 . . . . . . 7 {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ⊆ (Walks‘𝐺)
29 resmpt 6055 . . . . . . 7 ({𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ⊆ (Walks‘𝐺) → ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ↦ (2nd𝑡)))
3028, 29ax-mp 5 . . . . . 6 ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ↦ (2nd𝑡))
31 fveq2 6906 . . . . . . . 8 (𝑡 = 𝑝 → (2nd𝑡) = (2nd𝑝))
3231cbvmptv 5255 . . . . . . 7 (𝑡 ∈ (Walks‘𝐺) ↦ (2nd𝑡)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝))
3332reseq1i 5993 . . . . . 6 ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁})
3427, 30, 333eqtr2i 2771 . . . . 5 (𝑡𝑇 ↦ (2nd𝑡)) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁})
3534a1i 11 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑡𝑇 ↦ (2nd𝑡)) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}))
3625, 35eqtrid 2789 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹 = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}))
3726a1i 11 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁})
38 wlknwwlksnbij.w . . . 4 𝑊 = (𝑁 WWalksN 𝐺)
39 wwlksn 29857 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
4039adantl 481 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
4138, 40eqtrid 2789 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑊 = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
4236, 37, 41f1oeq123d 6842 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝐹:𝑇1-1-onto𝑊 ↔ ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}):{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}–1-1-onto→{𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)}))
4324, 42mpbird 257 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  wss 3951   class class class wbr 5143  cmpt 5225  cres 5687  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  cc 11153  1c1 11156   + caddc 11158  0cn0 12526  chash 14369  USPGraphcuspgr 29165  Walkscwlks 29614  WWalkscwwlks 29845   WWalksN cwwlksn 29846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-edg 29065  df-uhgr 29075  df-upgr 29099  df-uspgr 29167  df-wlks 29617  df-wwlks 29850  df-wwlksn 29851
This theorem is referenced by:  wlknwwlksnen  29909  wlksnwwlknvbij  29928
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