MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2clwwlkel Structured version   Visualization version   GIF version

Theorem 2clwwlkel 29335
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 29333 . . 3 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋})
32eleq2d 2824 . 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 6846 . . . 4 (𝑀 = π‘Š β†’ (π‘€β€˜(𝑁 βˆ’ 2)) = (π‘Šβ€˜(𝑁 βˆ’ 2)))
54eqeq1d 2739 . . 3 (𝑀 = π‘Š β†’ ((π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋 ↔ (π‘Šβ€˜(𝑁 βˆ’ 2)) = 𝑋))
65elrab 3650 . 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 397   = wceq 1542   ∈ wcel 2107  {crab 3410  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364   βˆ’ cmin 11392  2c2 12215  β„€β‰₯cuz 12770  ClWWalksNOncclwwlknon 29073
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367
This theorem is referenced by:  2clwwlk2clwwlk  29336  numclwwlk1lem2f1  29343
  Copyright terms: Public domain W3C validator