Theorem uvtxel1 29330

Description: Characterization of a universal vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 14-Feb-2022.)

Hypotheses

Ref	Expression
uvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)
isuvtx.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
uvtxel1	⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒))

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺,𝑘   𝑒,𝑉,𝑘   𝑒,𝑁,𝑘

Allowed substitution hint:   𝐸(𝑘)

Proof of Theorem uvtxel1

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4602	. . . 4 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁})
2	1	difeq2d 4092	. . 3 ⊢ (𝑛 = 𝑁 → (𝑉 ∖ {𝑛}) = (𝑉 ∖ {𝑁}))
3		preq2 4701	. . . . 5 ⊢ (𝑛 = 𝑁 → {𝑘, 𝑛} = {𝑘, 𝑁})
4	3	sseq1d 3981	. . . 4 ⊢ (𝑛 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑘, 𝑁} ⊆ 𝑒))
5	4	rexbidv 3158	. . 3 ⊢ (𝑛 = 𝑁 → (∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒))
6	2, 5	raleqbidv 3321	. 2 ⊢ (𝑛 = 𝑁 → (∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒))
7		uvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
8		isuvtx.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
9	7, 8	isuvtx 29329	. 2 ⊢ (UnivVtx‘𝐺) = {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒 ∈ 𝐸 {𝑘, 𝑛} ⊆ 𝑒}
10	6, 9	elrab2 3665	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒 ∈ 𝐸 {𝑘, 𝑁} ⊆ 𝑒))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ∖ cdif 3914   ⊆ wss 3917  {csn 4592  {cpr 4594  ‘cfv 6514  Vtxcvtx 28930  Edgcedg 28981  UnivVtxcuvtx 29319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-nbgr 29267  df-uvtx 29320

This theorem is referenced by: (None)