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Theorem clwlknf1oclwwlknlem2 28433
Description: Lemma 2 for clwlknf1oclwwlkn 28435: 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 2fveq3 6773 . . . 4 (𝑤 = 𝑐 → (♯‘(1st𝑤)) = (♯‘(1st𝑐)))
21eqeq1d 2740 . . 3 (𝑤 = 𝑐 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁))
32cbvrabv 3425 . 2 {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁}
4 nnge1 11990 . . . . 5 (𝑁 ∈ ℕ → 1 ≤ 𝑁)
5 breq2 5079 . . . . 5 ((♯‘(1st𝑐)) = 𝑁 → (1 ≤ (♯‘(1st𝑐)) ↔ 1 ≤ 𝑁))
64, 5syl5ibrcom 246 . . . 4 (𝑁 ∈ ℕ → ((♯‘(1st𝑐)) = 𝑁 → 1 ≤ (♯‘(1st𝑐))))
76pm4.71rd 563 . . 3 (𝑁 ∈ ℕ → ((♯‘(1st𝑐)) = 𝑁 ↔ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)))
87rabbidv 3413 . 2 (𝑁 ∈ ℕ → {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
93, 8eqtrid 2790 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  wa 396   = wceq 1539  wcel 2106  {crab 3068   class class class wbr 5075  cfv 6428  1st c1st 7820  1c1 10861  cle 10999  cn 11962  chash 14033  ClWalkscclwlks 28125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7580  ax-resscn 10917  ax-1cn 10918  ax-icn 10919  ax-addcl 10920  ax-addrcl 10921  ax-mulcl 10922  ax-mulrcl 10923  ax-mulcom 10924  ax-addass 10925  ax-mulass 10926  ax-distr 10927  ax-i2m1 10928  ax-1ne0 10929  ax-1rid 10930  ax-rnegex 10931  ax-rrecex 10932  ax-cnre 10933  ax-pre-lttri 10934  ax-pre-lttrn 10935  ax-pre-ltadd 10936  ax-pre-mulgt0 10937
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5486  df-eprel 5492  df-po 5500  df-so 5501  df-fr 5541  df-we 5543  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-pred 6197  df-ord 6264  df-on 6265  df-lim 6266  df-suc 6267  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-f1 6433  df-fo 6434  df-f1o 6435  df-fv 6436  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7705  df-2nd 7823  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-er 8487  df-en 8723  df-dom 8724  df-sdom 8725  df-pnf 11000  df-mnf 11001  df-xr 11002  df-ltxr 11003  df-le 11004  df-sub 11196  df-neg 11197  df-nn 11963
This theorem is referenced by:  clwlknf1oclwwlkn  28435
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