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Mirrors > Home > MPE Home > Th. List > isuvtx | Structured version Visualization version GIF version |
Description: The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Revised by AV, 14-Feb-2022.) |
Ref | Expression |
---|---|
uvtxel.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isuvtx.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
isuvtx | ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑣})∃𝑒 ∈ 𝐸 {𝑘, 𝑣} ⊆ 𝑒} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvtxel.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | uvtxval 29213 | . 2 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑣})𝑘 ∈ (𝐺 NeighbVtx 𝑣)} |
3 | isuvtx.e | . . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺) | |
4 | 1, 3 | nbgrel 29166 | . . . . . 6 ⊢ (𝑘 ∈ (𝐺 NeighbVtx 𝑣) ↔ ((𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉) ∧ 𝑘 ≠ 𝑣 ∧ ∃𝑒 ∈ 𝐸 {𝑣, 𝑘} ⊆ 𝑒)) |
5 | df-3an 1087 | . . . . . 6 ⊢ (((𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉) ∧ 𝑘 ≠ 𝑣 ∧ ∃𝑒 ∈ 𝐸 {𝑣, 𝑘} ⊆ 𝑒) ↔ (((𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉) ∧ 𝑘 ≠ 𝑣) ∧ ∃𝑒 ∈ 𝐸 {𝑣, 𝑘} ⊆ 𝑒)) | |
6 | 4, 5 | bitri 275 | . . . . 5 ⊢ (𝑘 ∈ (𝐺 NeighbVtx 𝑣) ↔ (((𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉) ∧ 𝑘 ≠ 𝑣) ∧ ∃𝑒 ∈ 𝐸 {𝑣, 𝑘} ⊆ 𝑒)) |
7 | prcom 4737 | . . . . . . . 8 ⊢ {𝑘, 𝑣} = {𝑣, 𝑘} | |
8 | 7 | sseq1i 4008 | . . . . . . 7 ⊢ ({𝑘, 𝑣} ⊆ 𝑒 ↔ {𝑣, 𝑘} ⊆ 𝑒) |
9 | 8 | rexbii 3091 | . . . . . 6 ⊢ (∃𝑒 ∈ 𝐸 {𝑘, 𝑣} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑣, 𝑘} ⊆ 𝑒) |
10 | id 22 | . . . . . . . . 9 ⊢ (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝑉) | |
11 | eldifi 4125 | . . . . . . . . 9 ⊢ (𝑘 ∈ (𝑉 ∖ {𝑣}) → 𝑘 ∈ 𝑉) | |
12 | 10, 11 | anim12ci 613 | . . . . . . . 8 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑘 ∈ (𝑉 ∖ {𝑣})) → (𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) |
13 | eldifsni 4794 | . . . . . . . . 9 ⊢ (𝑘 ∈ (𝑉 ∖ {𝑣}) → 𝑘 ≠ 𝑣) | |
14 | 13 | adantl 481 | . . . . . . . 8 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑘 ∈ (𝑉 ∖ {𝑣})) → 𝑘 ≠ 𝑣) |
15 | 12, 14 | jca 511 | . . . . . . 7 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑘 ∈ (𝑉 ∖ {𝑣})) → ((𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉) ∧ 𝑘 ≠ 𝑣)) |
16 | 15 | biantrurd 532 | . . . . . 6 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑘 ∈ (𝑉 ∖ {𝑣})) → (∃𝑒 ∈ 𝐸 {𝑣, 𝑘} ⊆ 𝑒 ↔ (((𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉) ∧ 𝑘 ≠ 𝑣) ∧ ∃𝑒 ∈ 𝐸 {𝑣, 𝑘} ⊆ 𝑒))) |
17 | 9, 16 | bitr2id 284 | . . . . 5 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑘 ∈ (𝑉 ∖ {𝑣})) → ((((𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉) ∧ 𝑘 ≠ 𝑣) ∧ ∃𝑒 ∈ 𝐸 {𝑣, 𝑘} ⊆ 𝑒) ↔ ∃𝑒 ∈ 𝐸 {𝑘, 𝑣} ⊆ 𝑒)) |
18 | 6, 17 | bitrid 283 | . . . 4 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑘 ∈ (𝑉 ∖ {𝑣})) → (𝑘 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∃𝑒 ∈ 𝐸 {𝑘, 𝑣} ⊆ 𝑒)) |
19 | 18 | ralbidva 3172 | . . 3 ⊢ (𝑣 ∈ 𝑉 → (∀𝑘 ∈ (𝑉 ∖ {𝑣})𝑘 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑣})∃𝑒 ∈ 𝐸 {𝑘, 𝑣} ⊆ 𝑒)) |
20 | 19 | rabbiia 3433 | . 2 ⊢ {𝑣 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑣})𝑘 ∈ (𝐺 NeighbVtx 𝑣)} = {𝑣 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑣})∃𝑒 ∈ 𝐸 {𝑘, 𝑣} ⊆ 𝑒} |
21 | 2, 20 | eqtri 2756 | 1 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑣})∃𝑒 ∈ 𝐸 {𝑘, 𝑣} ⊆ 𝑒} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 {crab 3429 ∖ cdif 3944 ⊆ wss 3947 {csn 4629 {cpr 4631 ‘cfv 6548 (class class class)co 7420 Vtxcvtx 28822 Edgcedg 28873 NeighbVtx cnbgr 29158 UnivVtxcuvtx 29211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-nbgr 29159 df-uvtx 29212 |
This theorem is referenced by: uvtxel1 29222 |
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