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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 7156 | . . 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 4804 | . 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 5349 | . 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 1536 ∈ wcel 2113 {crab 3141 〈cop 4566 ↦ cmpt 5139 ‘cfv 6348 (class class class)co 7149 ∈ cmpo 7151 1c1 10531 − cmin 10863 2c2 11686 ℤ≥cuz 12237 prefix cpfx 14027 Vtxcvtx 26779 ClWWalksNOncclwwlknon 27864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 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 7152 |
This theorem is referenced by: numclwwlk1lem2f1 28134 numclwwlk1lem2fo 28135 |
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