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Mirrors > Home > MPE Home > Th. List > clwwlknonmpo | Structured version Visualization version GIF version |
Description: (ClWWalksNOn‘𝐺) is an operator mapping a vertex 𝑣 and a nonnegative integer 𝑛 to the set of closed walks on 𝑣 of length 𝑛 as words over the set of vertices in a graph 𝐺. (Contributed by AV, 25-Feb-2022.) (Proof shortened by AV, 2-Mar-2024.) |
Ref | Expression |
---|---|
clwwlknonmpo | ⊢ (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6891 | . . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) | |
2 | eqidd 2732 | . . . 4 ⊢ (𝑔 = 𝐺 → ℕ0 = ℕ0) | |
3 | oveq2 7420 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑛 ClWWalksN 𝑔) = (𝑛 ClWWalksN 𝐺)) | |
4 | 3 | rabeqdv 3446 | . . . 4 ⊢ (𝑔 = 𝐺 → {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣} = {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) |
5 | 1, 2, 4 | mpoeq123dv 7487 | . . 3 ⊢ (𝑔 = 𝐺 → (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})) |
6 | df-clwwlknon 29774 | . . 3 ⊢ ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣})) | |
7 | fvex 6904 | . . . 4 ⊢ (Vtx‘𝐺) ∈ V | |
8 | nn0ex 12485 | . . . 4 ⊢ ℕ0 ∈ V | |
9 | 7, 8 | mpoex 8070 | . . 3 ⊢ (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ∈ V |
10 | 5, 6, 9 | fvmpt 6998 | . 2 ⊢ (𝐺 ∈ V → (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})) |
11 | fvprc 6883 | . . 3 ⊢ (¬ 𝐺 ∈ V → (ClWWalksNOn‘𝐺) = ∅) | |
12 | fvprc 6883 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅) | |
13 | 12 | orcd 870 | . . . 4 ⊢ (¬ 𝐺 ∈ V → ((Vtx‘𝐺) = ∅ ∨ ℕ0 = ∅)) |
14 | 0mpo0 7495 | . . . 4 ⊢ (((Vtx‘𝐺) = ∅ ∨ ℕ0 = ∅) → (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) = ∅) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) = ∅) |
16 | 11, 15 | eqtr4d 2774 | . 2 ⊢ (¬ 𝐺 ∈ V → (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})) |
17 | 10, 16 | pm2.61i 182 | 1 ⊢ (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 844 = wceq 1540 ∈ wcel 2105 {crab 3431 Vcvv 3473 ∅c0 4322 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 0cc0 11116 ℕ0cn0 12479 Vtxcvtx 28689 ClWWalksN cclwwlkn 29710 ClWWalksNOncclwwlknon 29773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-1cn 11174 ax-addcl 11176 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-nn 12220 df-n0 12480 df-clwwlknon 29774 |
This theorem is referenced by: clwwlknon 29776 clwwlk0on0 29778 clwwlknon0 29779 2clwwlk2clwwlklem 30032 |
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