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Theorem uvtxaval 26208
Description: The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.)
Hypothesis
Ref Expression
isuvtxa.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
uvtxaval (𝐺𝑊 → (UnivVtx‘𝐺) = {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
Distinct variable groups:   𝑛,𝐺,𝑣   𝑛,𝑉,𝑣
Allowed substitution hints:   𝑊(𝑣,𝑛)

Proof of Theorem uvtxaval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 df-uvtxa 26151 . . 3 UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
21a1i 11 . 2 (𝐺𝑊 → UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}))
3 fveq2 6158 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4 isuvtxa.v . . . . 5 𝑉 = (Vtx‘𝐺)
53, 4syl6eqr 2673 . . . 4 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
65difeq1d 3711 . . . . 5 (𝑔 = 𝐺 → ((Vtx‘𝑔) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
7 oveq1 6622 . . . . . 6 (𝑔 = 𝐺 → (𝑔 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑣))
87eleq2d 2684 . . . . 5 (𝑔 = 𝐺 → (𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
96, 8raleqbidv 3145 . . . 4 (𝑔 = 𝐺 → (∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
105, 9rabeqbidv 3185 . . 3 (𝑔 = 𝐺 → {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)} = {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
1110adantl 482 . 2 ((𝐺𝑊𝑔 = 𝐺) → {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)} = {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
12 elex 3202 . 2 (𝐺𝑊𝐺 ∈ V)
13 fvex 6168 . . . . 5 (Vtx‘𝐺) ∈ V
144, 13eqeltri 2694 . . . 4 𝑉 ∈ V
1514rabex 4783 . . 3 {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ∈ V
1615a1i 11 . 2 (𝐺𝑊 → {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ∈ V)
172, 11, 12, 16fvmptd 6255 1 (𝐺𝑊 → (UnivVtx‘𝐺) = {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2908  {crab 2912  Vcvv 3190  cdif 3557  {csn 4155  cmpt 4683  cfv 5857  (class class class)co 6615  Vtxcvtx 25808   NeighbVtx cnbgr 26145  UnivVtxcuvtxa 26146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-uvtxa 26151
This theorem is referenced by:  uvtxael  26209  uvtxaisvtx  26210  uvtxa0  26215  isuvtxa  26216  uvtxa01vtx0  26218  uvtxusgr  26224
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