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Theorem clwlknf1oclwwlknlem2 27253
Description: Lemma 2 for clwlknf1oclwwlkn 27255: The closed walks of a positive length are nonempty closed walks of this length. (Contributed by AV, 26-May-2022.)
Assertion
Ref Expression
clwlknf1oclwwlknlem2 (𝑁 ∈ ℕ → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
Distinct variable groups:   𝐺,𝑐,𝑤   𝑁,𝑐,𝑤

Proof of Theorem clwlknf1oclwwlknlem2
StepHypRef Expression
1 fveq2 6332 . . . . 5 (𝑤 = 𝑐 → (1st𝑤) = (1st𝑐))
21fveq2d 6336 . . . 4 (𝑤 = 𝑐 → (♯‘(1st𝑤)) = (♯‘(1st𝑐)))
32eqeq1d 2773 . . 3 (𝑤 = 𝑐 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁))
43cbvrabv 3349 . 2 {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁}
5 nnge1 11248 . . . . . 6 (𝑁 ∈ ℕ → 1 ≤ 𝑁)
6 breq2 4790 . . . . . 6 ((♯‘(1st𝑐)) = 𝑁 → (1 ≤ (♯‘(1st𝑐)) ↔ 1 ≤ 𝑁))
75, 6syl5ibrcom 237 . . . . 5 (𝑁 ∈ ℕ → ((♯‘(1st𝑐)) = 𝑁 → 1 ≤ (♯‘(1st𝑐))))
87pm4.71rd 552 . . . 4 (𝑁 ∈ ℕ → ((♯‘(1st𝑐)) = 𝑁 ↔ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)))
98adantr 466 . . 3 ((𝑁 ∈ ℕ ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1st𝑐)) = 𝑁 ↔ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)))
109rabbidva 3338 . 2 (𝑁 ∈ ℕ → {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
114, 10syl5eq 2817 1 (𝑁 ∈ ℕ → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {crab 3065   class class class wbr 4786  cfv 6031  1st c1st 7313  1c1 10139  cle 10277  cn 11222  chash 13321  ClWalkscclwlks 26901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223
This theorem is referenced by:  clwlknf1oclwwlkn  27255
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