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Theorem clwlknf1oclwwlknlem3 27981
 Description: Lemma 3 for clwlknf1oclwwlkn 27982: The bijective function of clwlknf1oclwwlkn 27982 is the bijective function of clwlkclwwlkf1o 27909 restricted to the closed walks with a fixed positive length. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)
Hypotheses
Ref Expression
clwlknf1oclwwlkn.a 𝐴 = (1st𝑐)
clwlknf1oclwwlkn.b 𝐵 = (2nd𝑐)
clwlknf1oclwwlkn.c 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
clwlknf1oclwwlkn.f 𝐹 = (𝑐𝐶 ↦ (𝐵 prefix (♯‘𝐴)))
Assertion
Ref Expression
clwlknf1oclwwlknlem3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶))
Distinct variable groups:   𝐶,𝑐   𝐺,𝑐,𝑤   𝑤,𝑁
Allowed substitution hints:   𝐴(𝑤,𝑐)   𝐵(𝑤,𝑐)   𝐶(𝑤)   𝐹(𝑤,𝑐)   𝑁(𝑐)

Proof of Theorem clwlknf1oclwwlknlem3
StepHypRef Expression
1 clwlknf1oclwwlkn.f . 2 𝐹 = (𝑐𝐶 ↦ (𝐵 prefix (♯‘𝐴)))
2 clwlknf1oclwwlkn.c . . . 4 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
3 nnge1 11715 . . . . . . 7 (𝑁 ∈ ℕ → 1 ≤ 𝑁)
4 breq2 5040 . . . . . . 7 ((♯‘(1st𝑤)) = 𝑁 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ 𝑁))
53, 4syl5ibrcom 250 . . . . . 6 (𝑁 ∈ ℕ → ((♯‘(1st𝑤)) = 𝑁 → 1 ≤ (♯‘(1st𝑤))))
65ad2antlr 726 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → ((♯‘(1st𝑤)) = 𝑁 → 1 ≤ (♯‘(1st𝑤))))
76ss2rabdv 3982 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))})
82, 7eqsstrid 3942 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 ⊆ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))})
98resmptd 5885 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶) = (𝑐𝐶 ↦ (𝐵 prefix (♯‘𝐴))))
101, 9eqtr4id 2812 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3074   class class class wbr 5036   ↦ cmpt 5116   ↾ cres 5530  ‘cfv 6340  (class class class)co 7156  1st c1st 7697  2nd c2nd 7698  1c1 10589   ≤ cle 10727  ℕcn 11687  ♯chash 13753   prefix cpfx 14092  USPGraphcuspgr 27054  ClWalkscclwlks 27672 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688 This theorem is referenced by:  clwlknf1oclwwlkn  27982
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