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Mirrors > Home > MPE Home > Th. List > numclwwlk1lem2fv | Structured version Visualization version GIF version |
Description: Value of the function 𝑇. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Oct-2022.) |
Ref | Expression |
---|---|
extwwlkfab.v | ⊢ 𝑉 = (Vtx‘𝐺) |
extwwlkfab.c | ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) |
extwwlkfab.f | ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) |
numclwwlk.t | ⊢ 𝑇 = (𝑢 ∈ (𝑋𝐶𝑁) ↦ 〈(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 − 1))〉) |
Ref | Expression |
---|---|
numclwwlk1lem2fv | ⊢ (𝑊 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑊) = 〈(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7438 | . . 3 ⊢ (𝑢 = 𝑊 → (𝑢 prefix (𝑁 − 2)) = (𝑊 prefix (𝑁 − 2))) | |
2 | fveq1 6906 | . . 3 ⊢ (𝑢 = 𝑊 → (𝑢‘(𝑁 − 1)) = (𝑊‘(𝑁 − 1))) | |
3 | 1, 2 | opeq12d 4886 | . 2 ⊢ (𝑢 = 𝑊 → 〈(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 − 1))〉 = 〈(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))〉) |
4 | numclwwlk.t | . 2 ⊢ 𝑇 = (𝑢 ∈ (𝑋𝐶𝑁) ↦ 〈(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 − 1))〉) | |
5 | opex 5475 | . 2 ⊢ 〈(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))〉 ∈ V | |
6 | 3, 4, 5 | fvmpt 7016 | 1 ⊢ (𝑊 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑊) = 〈(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 〈cop 4637 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1c1 11154 − cmin 11490 2c2 12319 ℤ≥cuz 12876 prefix cpfx 14705 Vtxcvtx 29028 ClWWalksNOncclwwlknon 30116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 |
This theorem is referenced by: numclwwlk1lem2f1 30386 numclwwlk1lem2fo 30387 |
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