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Theorem numclwwlk3lem2lem 29369
Description: Lemma for numclwwlk3lem2 29370: The set of closed vertices of a fixed length 𝑁 on a fixed vertex 𝑉 is the union of the set of closed walks of length 𝑁 at 𝑉 with the last but one vertex being 𝑉 and the set of closed walks of length 𝑁 at 𝑉 with the last but one vertex not being 𝑉. (Contributed by AV, 1-May-2022.)
Hypotheses
Ref Expression
numclwwlk3lem2.c 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
numclwwlk3lem2.h 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})
Assertion
Ref Expression
numclwwlk3lem2lem ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) = ((𝑋𝐻𝑁) βˆͺ (𝑋𝐢𝑁)))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑀
Allowed substitution hints:   𝐢(𝑀,𝑣,𝑛)   𝐻(𝑀,𝑣,𝑛)   𝑉(𝑀)

Proof of Theorem numclwwlk3lem2lem
StepHypRef Expression
1 numclwwlk3lem2.h . . . 4 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})
21numclwwlkovh0 29358 . . 3 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐻𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋})
3 numclwwlk3lem2.c . . . 4 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
432clwwlk 29333 . . 3 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋})
52, 4uneq12d 4129 . 2 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ ((𝑋𝐻𝑁) βˆͺ (𝑋𝐢𝑁)) = ({𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋} βˆͺ {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋}))
6 unrab 4270 . . 3 ({𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋} βˆͺ {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋}) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ ((π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋 ∨ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)}
7 exmidne 2954 . . . . . 6 ((π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋 ∨ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋)
8 orcom 869 . . . . . 6 (((π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋 ∨ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋) ↔ ((π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋 ∨ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋))
97, 8mpbir 230 . . . . 5 ((π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋 ∨ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)
109a1i 11 . . . 4 (𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) β†’ ((π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋 ∨ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋))
1110rabeqc 3422 . . 3 {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ ((π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋 ∨ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)} = (𝑋(ClWWalksNOnβ€˜πΊ)𝑁)
126, 11eqtri 2765 . 2 ({𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋} βˆͺ {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋}) = (𝑋(ClWWalksNOnβ€˜πΊ)𝑁)
135, 12eqtr2di 2794 1 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) = ((𝑋𝐻𝑁) βˆͺ (𝑋𝐢𝑁)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  {crab 3410   βˆͺ cun 3913  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:  numclwwlk3lem2  29370
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