Theorem uvtxval 29534

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.)

Hypothesis

Ref	Expression
uvtxval.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uvtxval	⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}

Distinct variable groups:   𝑛,𝐺,𝑣   𝑛,𝑉,𝑣

Proof of Theorem uvtxval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-uvtx 29533	. . 3 ⊢ UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
2		fveq2 6863	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
3		uvtxval.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
4	2, 3	eqtr4di 2814	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
5	4	difeq1d 4079	. . . 4 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
6		oveq1 7399	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑣))
7	6	eleq2d 2847	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
8	5, 7	raleqbidv 3335	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
9	1, 8	fvmptrabfv 7004	. 2 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ (Vtx‘𝐺) ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
10	3	eqcomi 2770	. . 3 ⊢ (Vtx‘𝐺) = 𝑉
11	10	rabeqi 3426	. 2 ⊢ {𝑣 ∈ (Vtx‘𝐺) ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
12	9, 11	eqtri 2784	1 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}

Syntax hints:   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413   ∖ cdif 3901  {csn 4581  ‘cfv 6517  (class class class)co 7392  Vtxcvtx 29143   NeighbVtx cnbgr 29479  UnivVtxcuvtx 29532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-uvtx 29533

This theorem is referenced by:  uvtxel  29535  uvtx0  29541  isuvtx  29542  uvtx01vtx  29544  uvtxusgr  29549