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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 7152 | . . 3 ⊢ (𝑢 = 𝑊 → (𝑢 prefix (𝑁 − 2)) = (𝑊 prefix (𝑁 − 2))) | |
2 | fveq1 6662 | . . 3 ⊢ (𝑢 = 𝑊 → (𝑢‘(𝑁 − 1)) = (𝑊‘(𝑁 − 1))) | |
3 | 1, 2 | opeq12d 4803 | . 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 5347 | . 2 ⊢ 〈(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))〉 ∈ V | |
6 | 3, 4, 5 | fvmpt 6761 | 1 ⊢ (𝑊 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑊) = 〈(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {crab 3139 〈cop 4563 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 1c1 10526 − cmin 10858 2c2 11680 ℤ≥cuz 12231 prefix cpfx 14020 Vtxcvtx 26708 ClWWalksNOncclwwlknon 27793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 |
This theorem is referenced by: numclwwlk1lem2f1 28063 numclwwlk1lem2fo 28064 |
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