Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 |
---|---|
01wlk.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
0wlkOns1 | ⊢ (𝑁 ∈ 𝑉 → ∅(𝑁(WalksOn‘𝐺)𝑁)〈“𝑁”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1val 13231 | . . 3 ⊢ (𝑁 ∈ 𝑉 → 〈“𝑁”〉 = {〈0, 𝑁〉}) | |
2 | 0z 11265 | . . . . . 6 ⊢ 0 ∈ ℤ | |
3 | 2 | jctl 562 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → (0 ∈ ℤ ∧ 𝑁 ∈ 𝑉)) |
4 | f1sng 6090 | . . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ 𝑉) → {〈0, 𝑁〉}:{0}–1-1→𝑉) | |
5 | f1f 6014 | . . . . 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 12254 | . . . . . 6 ⊢ (0 ∈ ℤ → (0...0) = {0}) | |
9 | 2, 8 | mp1i 13 | . . . . 5 ⊢ (〈“𝑁”〉 = {〈0, 𝑁〉} → (0...0) = {0}) |
10 | 7, 9 | feq12d 5946 | . . . 4 ⊢ (〈“𝑁”〉 = {〈0, 𝑁〉} → (〈“𝑁”〉:(0...0)⟶𝑉 ↔ {〈0, 𝑁〉}:{0}⟶𝑉)) |
11 | 6, 10 | syl5ibrcom 236 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (〈“𝑁”〉 = {〈0, 𝑁〉} → 〈“𝑁”〉:(0...0)⟶𝑉)) |
12 | 1, 11 | mpd 15 | . 2 ⊢ (𝑁 ∈ 𝑉 → 〈“𝑁”〉:(0...0)⟶𝑉) |
13 | s1fv 13243 | . 2 ⊢ (𝑁 ∈ 𝑉 → (〈“𝑁”〉‘0) = 𝑁) | |
14 | 01wlk.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
15 | 14 | 0wlkOn 41288 | . 2 ⊢ ((〈“𝑁”〉:(0...0)⟶𝑉 ∧ (〈“𝑁”〉‘0) = 𝑁) → ∅(𝑁(WalksOn‘𝐺)𝑁)〈“𝑁”〉) |
16 | 12, 13, 15 | syl2anc 691 | 1 ⊢ (𝑁 ∈ 𝑉 → ∅(𝑁(WalksOn‘𝐺)𝑁)〈“𝑁”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 {csn 4125 〈cop 4131 class class class wbr 4583 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℤcz 11254 ...cfz 12197 〈“cs1 13149 Vtxcvtx 25673 WalksOncwlkson 40798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-s1 13157 df-1wlks 40800 df-wlkson 40802 |
This theorem is referenced by: (None) |
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