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Mirrors > Home > MPE Home > Th. List > uvtxval | 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, 29-Oct-2020.) (Revised by AV, 14-Feb-2022.) |
Ref | Expression |
---|---|
uvtxval.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
uvtxval | ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-uvtx 27753 | . . 3 ⊢ UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}) | |
2 | fveq2 6774 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) | |
3 | uvtxval.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 2, 3 | eqtr4di 2796 | . . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
5 | 4 | difeq1d 4056 | . . . 4 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∖ {𝑣}) = (𝑉 ∖ {𝑣})) |
6 | oveq1 7282 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑣)) | |
7 | 6 | eleq2d 2824 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑣))) |
8 | 5, 7 | raleqbidv 3336 | . . 3 ⊢ (𝑔 = 𝐺 → (∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))) |
9 | 1, 8 | fvmptrabfv 6906 | . 2 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ (Vtx‘𝐺) ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} |
10 | 3 | eqcomi 2747 | . . 3 ⊢ (Vtx‘𝐺) = 𝑉 |
11 | 10 | rabeqi 3416 | . 2 ⊢ {𝑣 ∈ (Vtx‘𝐺) ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} |
12 | 9, 11 | eqtri 2766 | 1 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 ∖ cdif 3884 {csn 4561 ‘cfv 6433 (class class class)co 7275 Vtxcvtx 27366 NeighbVtx cnbgr 27699 UnivVtxcuvtx 27752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-uvtx 27753 |
This theorem is referenced by: uvtxel 27755 uvtx0 27761 isuvtx 27762 uvtx01vtx 27764 uvtxusgr 27769 |
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