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Theorem wlknwwlksnbij 29692
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 2727 . . 3 (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝))
21wlkswwlksf1o 29683 . . . 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 6900 . . . . 5 (𝑞 = (2nd𝑝) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd𝑝)) = (𝑁 + 1)))
543ad2ant3 1133 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd𝑝)) = (𝑁 + 1)))
6 wlkcpr 29436 . . . . . . 7 (𝑝 ∈ (Walks‘𝐺) ↔ (1st𝑝)(Walks‘𝐺)(2nd𝑝))
7 wlklenvp1 29425 . . . . . . . 8 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → (♯‘(2nd𝑝)) = ((♯‘(1st𝑝)) + 1))
8 eqeq1 2731 . . . . . . . . . 10 ((♯‘(2nd𝑝)) = ((♯‘(1st𝑝)) + 1) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ ((♯‘(1st𝑝)) + 1) = (𝑁 + 1)))
9 wlkcl 29422 . . . . . . . . . . . . 13 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → (♯‘(1st𝑝)) ∈ ℕ0)
109nn0cnd 12558 . . . . . . . . . . . 12 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → (♯‘(1st𝑝)) ∈ ℂ)
1110adantr 480 . . . . . . . . . . 11 (((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (♯‘(1st𝑝)) ∈ ℂ)
12 nn0cn 12506 . . . . . . . . . . . . 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 11233 . . . . . . . . . . 11 (((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 1 ∈ ℂ)
1611, 14, 15addcan2d 11442 . . . . . . . . . 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 216 . . . . . 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 1130 . . . 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 7130 . 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 4073 . . . . . . 7 {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ⊆ (Walks‘𝐺)
29 resmpt 6035 . . . . . . 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 6891 . . . . . . . 8 (𝑡 = 𝑝 → (2nd𝑡) = (2nd𝑝))
3231cbvmptv 5255 . . . . . . 7 (𝑡 ∈ (Walks‘𝐺) ↦ (2nd𝑡)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝))
3332reseq1i 5975 . . . . . 6 ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁})
3427, 30, 333eqtr2i 2761 . . . . 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 2779 . . 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 29641 . . . . 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 2779 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑊 = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
4236, 37, 41f1oeq123d 6827 . 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 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  {crab 3427  wss 3944   class class class wbr 5142  cmpt 5225  cres 5674  1-1-ontowf1o 6541  cfv 6542  (class class class)co 7414  1st c1st 7985  2nd c2nd 7986  cc 11130  1c1 11133   + caddc 11135  0cn0 12496  chash 14315  USPGraphcuspgr 28954  Walkscwlks 29403  WWalkscwwlks 29629   WWalksN cwwlksn 29630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-map 8840  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-dju 9918  df-card 9956  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-nn 12237  df-2 12299  df-n0 12497  df-xnn0 12569  df-z 12583  df-uz 12847  df-fz 13511  df-fzo 13654  df-hash 14316  df-word 14491  df-edg 28854  df-uhgr 28864  df-upgr 28888  df-uspgr 28956  df-wlks 29406  df-wwlks 29634  df-wwlksn 29635
This theorem is referenced by:  wlknwwlksnen  29693  wlksnwwlknvbij  29712
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