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Theorem numclwwlkovh0 30169
Description: Value of operation 𝐻, mapping a vertex 𝑣 and an integer 𝑛 greater than 1 to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v" according to definition 7 in [Huneke] p. 2. (Contributed by AV, 1-May-2022.)
Hypothesis
Ref Expression
numclwwlkovh.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
numclwwlkovh0 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐻𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋})
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑀
Allowed substitution hints:   𝐻(𝑀,𝑣,𝑛)   𝑉(𝑀)

Proof of Theorem numclwwlkovh0
StepHypRef Expression
1 oveq12 7423 . . 3 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) = (𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
2 oveq1 7421 . . . . . 6 (𝑛 = 𝑁 β†’ (𝑛 βˆ’ 2) = (𝑁 βˆ’ 2))
32adantl 481 . . . . 5 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ (𝑛 βˆ’ 2) = (𝑁 βˆ’ 2))
43fveq2d 6895 . . . 4 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ (π‘€β€˜(𝑛 βˆ’ 2)) = (π‘€β€˜(𝑁 βˆ’ 2)))
5 simpl 482 . . . 4 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ 𝑣 = 𝑋)
64, 5neeq12d 2997 . . 3 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ ((π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣 ↔ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋))
71, 6rabeqbidv 3444 . 2 ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) β†’ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣} = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋})
8 numclwwlkovh.h . 2 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})
9 ovex 7447 . . 3 (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∈ V
109rabex 5328 . 2 {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋} ∈ V
117, 8, 10ovmpoa 7570 1 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐻𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  {crab 3427  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416   βˆ’ cmin 11466  2c2 12289  β„€β‰₯cuz 12844  ClWWalksNOncclwwlknon 29884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419
This theorem is referenced by:  numclwwlkovh  30170  numclwwlk3lem2lem  30180  numclwwlk3lem2  30181
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