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Theorem wlknwwlksnbij 29918
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 2735 . . 3 (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝))
21wlkswwlksf1o 29909 . . . 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 6916 . . . . 5 (𝑞 = (2nd𝑝) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd𝑝)) = (𝑁 + 1)))
543ad2ant3 1134 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd𝑝)) = (𝑁 + 1)))
6 wlkcpr 29662 . . . . . . 7 (𝑝 ∈ (Walks‘𝐺) ↔ (1st𝑝)(Walks‘𝐺)(2nd𝑝))
7 wlklenvp1 29651 . . . . . . . 8 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → (♯‘(2nd𝑝)) = ((♯‘(1st𝑝)) + 1))
8 eqeq1 2739 . . . . . . . . . 10 ((♯‘(2nd𝑝)) = ((♯‘(1st𝑝)) + 1) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ ((♯‘(1st𝑝)) + 1) = (𝑁 + 1)))
9 wlkcl 29648 . . . . . . . . . . . . 13 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → (♯‘(1st𝑝)) ∈ ℕ0)
109nn0cnd 12587 . . . . . . . . . . . 12 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → (♯‘(1st𝑝)) ∈ ℂ)
1110adantr 480 . . . . . . . . . . 11 (((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (♯‘(1st𝑝)) ∈ ℂ)
12 nn0cn 12534 . . . . . . . . . . . . 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 11254 . . . . . . . . . . 11 (((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 1 ∈ ℂ)
1611, 14, 15addcan2d 11463 . . . . . . . . . 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 1131 . . . 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 5244 . . . . . 6 (𝑡𝑇 ↦ (2nd𝑡)) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ↦ (2nd𝑡))
28 ssrab2 4090 . . . . . . 7 {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ⊆ (Walks‘𝐺)
29 resmpt 6057 . . . . . . 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 6907 . . . . . . . 8 (𝑡 = 𝑝 → (2nd𝑡) = (2nd𝑝))
3231cbvmptv 5261 . . . . . . 7 (𝑡 ∈ (Walks‘𝐺) ↦ (2nd𝑡)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝))
3332reseq1i 5996 . . . . . 6 ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁})
3427, 30, 333eqtr2i 2769 . . . . 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 2787 . . 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 29867 . . . . 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 2787 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑊 = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
4236, 37, 41f1oeq123d 6843 . 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 1086   = wceq 1537  wcel 2106  {crab 3433  wss 3963   class class class wbr 5148  cmpt 5231  cres 5691  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  cc 11151  1c1 11154   + caddc 11156  0cn0 12524  chash 14366  USPGraphcuspgr 29180  Walkscwlks 29629  WWalkscwwlks 29855   WWalksN cwwlksn 29856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-edg 29080  df-uhgr 29090  df-upgr 29114  df-uspgr 29182  df-wlks 29632  df-wwlks 29860  df-wwlksn 29861
This theorem is referenced by:  wlknwwlksnen  29919  wlksnwwlknvbij  29938
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