Theorem uvtxaval 26287

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	⊢ (𝐺 ∈ 𝑊 → (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀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.

Step	Hyp	Ref	Expression
1		df-uvtxa 26230	. . 3 ⊢ UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
2	1	a1i 11	. 2 ⊢ (𝐺 ∈ 𝑊 → UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}))
3		fveq2 6191	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4		isuvtxa.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5	3, 4	syl6eqr 2674	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
6	5	difeq1d 3727	. . . . 5 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
7		oveq1 6657	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑣))
8	7	eleq2d 2687	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
9	6, 8	raleqbidv 3152	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
10	5, 9	rabeqbidv 3195	. . 3 ⊢ (𝑔 = 𝐺 → {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)} = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
11	10	adantl 482	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑔 = 𝐺) → {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)} = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
12		elex 3212	. 2 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
13		fvex 6201	. . . . 5 ⊢ (Vtx‘𝐺) ∈ V
14	4, 13	eqeltri 2697	. . . 4 ⊢ 𝑉 ∈ V
15	14	rabex 4813	. . 3 ⊢ {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ∈ V
16	15	a1i 11	. 2 ⊢ (𝐺 ∈ 𝑊 → {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ∈ V)
17	2, 11, 12, 16	fvmptd 6288	1 ⊢ (𝐺 ∈ 𝑊 → (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571  {csn 4177   ↦ cmpt 4729  ‘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:  uvtxael  26288  uvtxaisvtx  26289  uvtxa0  26294  isuvtxa  26295  uvtxa01vtx0  26297  uvtxusgr  26303