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Mirrors > Home > MPE Home > Th. List > uvtxael | Structured version Visualization version GIF version |
Description: A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.) |
Ref | Expression |
---|---|
uvtxael.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
uvtxael | ⊢ (𝐺 ∈ 𝑊 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvtxael.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | uvtxaval 26287 | . . 3 ⊢ (𝐺 ∈ 𝑊 → (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}) |
3 | 2 | eleq2d 2687 | . 2 ⊢ (𝐺 ∈ 𝑊 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ 𝑁 ∈ {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})) |
4 | sneq 4187 | . . . . 5 ⊢ (𝑣 = 𝑁 → {𝑣} = {𝑁}) | |
5 | 4 | difeq2d 3728 | . . . 4 ⊢ (𝑣 = 𝑁 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑁})) |
6 | oveq2 6658 | . . . . 5 ⊢ (𝑣 = 𝑁 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑁)) | |
7 | 6 | eleq2d 2687 | . . . 4 ⊢ (𝑣 = 𝑁 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑁))) |
8 | 5, 7 | raleqbidv 3152 | . . 3 ⊢ (𝑣 = 𝑁 → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))) |
9 | 8 | elrab 3363 | . 2 ⊢ (𝑁 ∈ {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))) |
10 | 3, 9 | syl6bb 276 | 1 ⊢ (𝐺 ∈ 𝑊 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 ∖ cdif 3571 {csn 4177 ‘cfv 5888 (class class class)co 6650 Vtxcvtx 25874 NeighbVtx cnbgr 26224 UnivVtxcuvtxa 26225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-uvtxa 26230 |
This theorem is referenced by: vtxnbuvtx 26291 uvtx2vtx1edg 26299 uvtx2vtx1edgb 26300 uvtxnbgrb 26302 iscplgrnb 26312 cplgr1v 26326 cusgrexi 26339 structtocusgr 26342 |
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