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Mirrors > Home > MPE Home > Th. List > 0wlkons1 | Structured version Visualization version GIF version |
Description: A walk of length 0 from a vertex to itself. (Contributed by AV, 17-Apr-2021.) |
Ref | Expression |
---|---|
0wlk.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
0wlkons1 | ⊢ (𝑁 ∈ 𝑉 → ∅(𝑁(WalksOn‘𝐺)𝑁)〈“𝑁”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1val 14231 | . . 3 ⊢ (𝑁 ∈ 𝑉 → 〈“𝑁”〉 = {〈0, 𝑁〉}) | |
2 | 0z 12260 | . . . . . 6 ⊢ 0 ∈ ℤ | |
3 | 2 | jctl 523 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → (0 ∈ ℤ ∧ 𝑁 ∈ 𝑉)) |
4 | f1sng 6741 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ 𝑉) → {〈0, 𝑁〉}:{0}–1-1→𝑉) | |
5 | f1f 6654 | . . . . 5 ⊢ ({〈0, 𝑁〉}:{0}–1-1→𝑉 → {〈0, 𝑁〉}:{0}⟶𝑉) | |
6 | 3, 4, 5 | 3syl 18 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → {〈0, 𝑁〉}:{0}⟶𝑉) |
7 | id 22 | . . . . 5 ⊢ (〈“𝑁”〉 = {〈0, 𝑁〉} → 〈“𝑁”〉 = {〈0, 𝑁〉}) | |
8 | fzsn 13227 | . . . . . 6 ⊢ (0 ∈ ℤ → (0...0) = {0}) | |
9 | 2, 8 | mp1i 13 | . . . . 5 ⊢ (〈“𝑁”〉 = {〈0, 𝑁〉} → (0...0) = {0}) |
10 | 7, 9 | feq12d 6572 | . . . 4 ⊢ (〈“𝑁”〉 = {〈0, 𝑁〉} → (〈“𝑁”〉:(0...0)⟶𝑉 ↔ {〈0, 𝑁〉}:{0}⟶𝑉)) |
11 | 6, 10 | syl5ibrcom 246 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (〈“𝑁”〉 = {〈0, 𝑁〉} → 〈“𝑁”〉:(0...0)⟶𝑉)) |
12 | 1, 11 | mpd 15 | . 2 ⊢ (𝑁 ∈ 𝑉 → 〈“𝑁”〉:(0...0)⟶𝑉) |
13 | s1fv 14243 | . 2 ⊢ (𝑁 ∈ 𝑉 → (〈“𝑁”〉‘0) = 𝑁) | |
14 | 0wlk.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
15 | 14 | 0wlkon 28385 | . 2 ⊢ ((〈“𝑁”〉:(0...0)⟶𝑉 ∧ (〈“𝑁”〉‘0) = 𝑁) → ∅(𝑁(WalksOn‘𝐺)𝑁)〈“𝑁”〉) |
16 | 12, 13, 15 | syl2anc 583 | 1 ⊢ (𝑁 ∈ 𝑉 → ∅(𝑁(WalksOn‘𝐺)𝑁)〈“𝑁”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∅c0 4253 {csn 4558 〈cop 4564 class class class wbr 5070 ⟶wf 6414 –1-1→wf1 6415 ‘cfv 6418 (class class class)co 7255 0cc0 10802 ℤcz 12249 ...cfz 13168 〈“cs1 14228 Vtxcvtx 27269 WalksOncwlkson 27867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-hash 13973 df-word 14146 df-s1 14229 df-wlks 27869 df-wlkson 27870 |
This theorem is referenced by: (None) |
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