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Theorem 2clwwlkel 30111
Description: Characterization of an element of the value of operation 𝐢, i.e., of a word being a double loop of length 𝑁 on vertex 𝑋. (Contributed by Alexander van der Vekens, 24-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 20-Apr-2022.)
Hypothesis
Ref Expression
2clwwlk.c 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
Assertion
Ref Expression
2clwwlkel ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (π‘Š ∈ (𝑋𝐢𝑁) ↔ (π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = 𝑋)))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑀   𝑀,π‘Š
Allowed substitution hints:   𝐢(𝑀,𝑣,𝑛)   𝑉(𝑀)   π‘Š(𝑣,𝑛)

Proof of Theorem 2clwwlkel
StepHypRef Expression
1 2clwwlk.c . . . 4 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
212clwwlk 30109 . . 3 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋})
32eleq2d 2813 . 2 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (π‘Š ∈ (𝑋𝐢𝑁) ↔ π‘Š ∈ {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋}))
4 fveq1 6884 . . . 4 (𝑀 = π‘Š β†’ (π‘€β€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜(𝑁 βˆ’ 2)))
54eqeq1d 2728 . . 3 (𝑀 = π‘Š β†’ ((π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋 ↔ (π‘Šβ€˜(𝑁 βˆ’ 2)) = 𝑋))
65elrab 3678 . 2 (π‘Š ∈ {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋} ↔ (π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = 𝑋))
73, 6bitrdi 287 1 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (π‘Š ∈ (𝑋𝐢𝑁) ↔ (π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = 𝑋)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3426  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407   βˆ’ cmin 11448  2c2 12271  β„€β‰₯cuz 12826  ClWWalksNOncclwwlknon 29849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410
This theorem is referenced by:  2clwwlk2clwwlk  30112  numclwwlk1lem2f1  30119
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