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Mirrors > Home > MPE Home > Th. List > nbusgrf1o1 | Structured version Visualization version GIF version |
Description: The set of neighbors of a vertex is isomorphic to the set of edges containing the vertex in a simple graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.) |
Ref | Expression |
---|---|
nbusgrf1o1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
nbusgrf1o1.e | ⊢ 𝐸 = (Edg‘𝐺) |
nbusgrf1o1.n | ⊢ 𝑁 = (𝐺 NeighbVtx 𝑈) |
nbusgrf1o1.i | ⊢ 𝐼 = {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} |
Ref | Expression |
---|---|
nbusgrf1o1 | ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ∃𝑓 𝑓:𝑁–1-1-onto→𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbusgrf1o1.n | . . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑈) | |
2 | 1 | ovexi 7450 | . . 3 ⊢ 𝑁 ∈ V |
3 | mptexg 7229 | . . 3 ⊢ (𝑁 ∈ V → (𝑛 ∈ 𝑁 ↦ {𝑈, 𝑛}) ∈ V) | |
4 | 2, 3 | mp1i 13 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝑛 ∈ 𝑁 ↦ {𝑈, 𝑛}) ∈ V) |
5 | nbusgrf1o1.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
6 | nbusgrf1o1.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
7 | nbusgrf1o1.i | . . 3 ⊢ 𝐼 = {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} | |
8 | eqid 2725 | . . 3 ⊢ (𝑛 ∈ 𝑁 ↦ {𝑈, 𝑛}) = (𝑛 ∈ 𝑁 ↦ {𝑈, 𝑛}) | |
9 | 5, 6, 1, 7, 8 | nbusgrf1o0 29226 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝑛 ∈ 𝑁 ↦ {𝑈, 𝑛}):𝑁–1-1-onto→𝐼) |
10 | f1oeq1 6822 | . 2 ⊢ (𝑓 = (𝑛 ∈ 𝑁 ↦ {𝑈, 𝑛}) → (𝑓:𝑁–1-1-onto→𝐼 ↔ (𝑛 ∈ 𝑁 ↦ {𝑈, 𝑛}):𝑁–1-1-onto→𝐼)) | |
11 | 4, 9, 10 | spcedv 3577 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ∃𝑓 𝑓:𝑁–1-1-onto→𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {crab 3419 Vcvv 3463 {cpr 4626 ↦ cmpt 5226 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7416 Vtxcvtx 28853 Edgcedg 28904 USGraphcusgr 29006 NeighbVtx cnbgr 29189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13517 df-hash 14322 df-edg 28905 df-upgr 28939 df-umgr 28940 df-uspgr 29007 df-usgr 29008 df-nbgr 29190 |
This theorem is referenced by: nbusgrf1o 29228 |
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