Theorem uvtxavalOLD 26513

Description: Obsolete version of uvtxel 26514 as of 14-Feb-2022. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
uvtxavalOLD	⊢ (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 uvtxavalOLD

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-uvtx 26511	. . 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 6333	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4		uvtxval.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
5	3, 4	syl6eqr 2823	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
6	5	difeq1d 3878	. . . . 5 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
7		oveq1 6803	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑣))
8	7	eleq2d 2836	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
9	6, 8	raleqbidv 3301	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
10	5, 9	rabeqbidv 3345	. . 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 467	. 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 3364	. 2 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
13		fvex 6344	. . . . 5 ⊢ (Vtx‘𝐺) ∈ V
14	4, 13	eqeltri 2846	. . . 4 ⊢ 𝑉 ∈ V
15	14	rabex 4947	. . 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 6432	1 ⊢ (𝐺 ∈ 𝑊 → (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})

Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  ∀wral 3061  {crab 3065  Vcvv 3351   ∖ cdif 3720  {csn 4317   ↦ cmpt 4864  ‘cfv 6030  (class class class)co 6796  Vtxcvtx 26095   NeighbVtx cnbgr 26447  UnivVtxcuvtx 26510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-uvtx 26511

This theorem is referenced by:  uvtxaelOLD  26515  isuvtxaOLD  26523  uvtxa01vtx0OLD  26527