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Mirrors > Home > MPE Home > Th. List > fusgreg2wsp | Structured version Visualization version GIF version |
Description: In a finite simple graph, the set of all paths of length 2 is the union of all the paths of length 2 over the vertices which are in the middle of such a path. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 10-Jan-2022.) |
Ref | Expression |
---|---|
frgrhash2wsp.v | ⊢ 𝑉 = (Vtx‘𝐺) |
fusgreg2wsp.m | ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) |
Ref | Expression |
---|---|
fusgreg2wsp | ⊢ (𝐺 ∈ FinUSGraph → (2 WSPathsN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wspthsswwlkn 29947 | . . . . . . . 8 ⊢ (2 WSPathsN 𝐺) ⊆ (2 WWalksN 𝐺) | |
2 | 1 | sseli 3990 | . . . . . . 7 ⊢ (𝑝 ∈ (2 WSPathsN 𝐺) → 𝑝 ∈ (2 WWalksN 𝐺)) |
3 | frgrhash2wsp.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 3 | midwwlks2s3 29981 | . . . . . . 7 ⊢ (𝑝 ∈ (2 WWalksN 𝐺) → ∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥) |
5 | 2, 4 | syl 17 | . . . . . 6 ⊢ (𝑝 ∈ (2 WSPathsN 𝐺) → ∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥) |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) → ∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥)) |
7 | 6 | pm4.71rd 562 | . . . 4 ⊢ (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) ↔ (∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺)))) |
8 | ancom 460 | . . . . . . 7 ⊢ ((𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ ((𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺))) | |
9 | 8 | rexbii 3091 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝑉 (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ ∃𝑥 ∈ 𝑉 ((𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺))) |
10 | r19.41v 3186 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝑉 ((𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺)) ↔ (∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺))) | |
11 | 9, 10 | bitr2i 276 | . . . . 5 ⊢ ((∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺)) ↔ ∃𝑥 ∈ 𝑉 (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥)) |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝐺 ∈ FinUSGraph → ((∃𝑥 ∈ 𝑉 (𝑝‘1) = 𝑥 ∧ 𝑝 ∈ (2 WSPathsN 𝐺)) ↔ ∃𝑥 ∈ 𝑉 (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥))) |
13 | fusgreg2wsp.m | . . . . . . . 8 ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) | |
14 | 3, 13 | fusgreg2wsplem 30361 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑥) ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥))) |
15 | 14 | bicomd 223 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 → ((𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ 𝑝 ∈ (𝑀‘𝑥))) |
16 | 15 | adantl 481 | . . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑥 ∈ 𝑉) → ((𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ 𝑝 ∈ (𝑀‘𝑥))) |
17 | 16 | rexbidva 3174 | . . . 4 ⊢ (𝐺 ∈ FinUSGraph → (∃𝑥 ∈ 𝑉 (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑥) ↔ ∃𝑥 ∈ 𝑉 𝑝 ∈ (𝑀‘𝑥))) |
18 | 7, 12, 17 | 3bitrd 305 | . . 3 ⊢ (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥 ∈ 𝑉 𝑝 ∈ (𝑀‘𝑥))) |
19 | eliun 4999 | . . 3 ⊢ (𝑝 ∈ ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥) ↔ ∃𝑥 ∈ 𝑉 𝑝 ∈ (𝑀‘𝑥)) | |
20 | 18, 19 | bitr4di 289 | . 2 ⊢ (𝐺 ∈ FinUSGraph → (𝑝 ∈ (2 WSPathsN 𝐺) ↔ 𝑝 ∈ ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥))) |
21 | 20 | eqrdv 2732 | 1 ⊢ (𝐺 ∈ FinUSGraph → (2 WSPathsN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 {crab 3432 ∪ ciun 4995 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 1c1 11153 2c2 12318 Vtxcvtx 29027 FinUSGraphcfusgr 29347 WWalksN cwwlksn 29855 WSPathsN cwwspthsn 29857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-s2 14883 df-s3 14884 df-wwlks 29859 df-wwlksn 29860 df-wspthsn 29862 |
This theorem is referenced by: fusgreghash2wsp 30366 |
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