Theorem uvtxel 26017

Description: A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)

Assertion

Ref	Expression
uvtxel	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁}){𝑘, 𝑁} ∈ ran 𝐸)))

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

Allowed substitution hints:   𝑋(𝑘)   𝑌(𝑘)

Proof of Theorem uvtxel

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isuvtx 26016	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 UnivVertex 𝐸) = {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸})
2	1	eleq2d 2673	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸}))
3		sneq 4135	. . . . 5 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁})
4	3	difeq2d 3690	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑉 ∖ {𝑛}) = (𝑉 ∖ {𝑁}))
5		preq2 4213	. . . . 5 ⊢ (𝑛 = 𝑁 → {𝑘, 𝑛} = {𝑘, 𝑁})
6	5	eleq1d 2672	. . . 4 ⊢ (𝑛 = 𝑁 → ({𝑘, 𝑛} ∈ ran 𝐸 ↔ {𝑘, 𝑁} ∈ ran 𝐸))
7	4, 6	raleqbidv 3129	. . 3 ⊢ (𝑛 = 𝑁 → (∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑁}){𝑘, 𝑁} ∈ ran 𝐸))
8	7	elrab 3331	. 2 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸} ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁}){𝑘, 𝑁} ∈ ran 𝐸))
9	2, 8	syl6bb 275	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁}){𝑘, 𝑁} ∈ ran 𝐸)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∖ cdif 3537  {csn 4125  {cpr 4127  ran crn 5039  (class class class)co 6549   UnivVertex cuvtx 25948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-uvtx 25951

This theorem is referenced by:  uvtxnbgra  26021  uvtxnb  26025