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Mirrors > Home > MPE Home > Th. List > 0pthon1 | Structured version Visualization version GIF version |
Description: A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.) |
Ref | Expression |
---|---|
0pthon.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
0pthon1 | ⊢ (𝑁 ∈ 𝑉 → ∅(𝑁(PathsOn‘𝐺)𝑁){〈0, 𝑁〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2740 | . . 3 ⊢ (𝑁 ∈ 𝑉 → {〈0, 𝑁〉} = {〈0, 𝑁〉}) | |
2 | 1fv 13120 | . . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ {〈0, 𝑁〉} = {〈0, 𝑁〉}) → ({〈0, 𝑁〉}:(0...0)⟶𝑉 ∧ ({〈0, 𝑁〉}‘0) = 𝑁)) | |
3 | 1, 2 | mpdan 687 | . 2 ⊢ (𝑁 ∈ 𝑉 → ({〈0, 𝑁〉}:(0...0)⟶𝑉 ∧ ({〈0, 𝑁〉}‘0) = 𝑁)) |
4 | 0pthon.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | 4 | 0pthon 28067 | . 2 ⊢ (({〈0, 𝑁〉}:(0...0)⟶𝑉 ∧ ({〈0, 𝑁〉}‘0) = 𝑁) → ∅(𝑁(PathsOn‘𝐺)𝑁){〈0, 𝑁〉}) |
6 | 3, 5 | syl 17 | 1 ⊢ (𝑁 ∈ 𝑉 → ∅(𝑁(PathsOn‘𝐺)𝑁){〈0, 𝑁〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∅c0 4212 {csn 4517 〈cop 4523 class class class wbr 5031 ⟶wf 6336 ‘cfv 6340 (class class class)co 7173 0cc0 10618 ...cfz 12984 Vtxcvtx 26944 PathsOncpthson 27658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-pm 8443 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-n0 11980 df-z 12066 df-uz 12328 df-fz 12985 df-fzo 13128 df-hash 13786 df-word 13959 df-wlks 27544 df-wlkson 27545 df-trls 27637 df-trlson 27638 df-pths 27660 df-pthson 27662 |
This theorem is referenced by: 0pthonv 28069 |
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