Theorem isuvtx 26016

Description: The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)

Assertion

Ref	Expression
isuvtx	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 UnivVertex 𝐸) = {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸})

Distinct variable groups:   𝑘,𝑉,𝑛   𝑘,𝐸,𝑛   𝑛,𝑋   𝑛,𝑌

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

Proof of Theorem isuvtx

Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-uvtx 25951	. 2 ⊢ UnivVertex = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑛 ∈ 𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒})
2		elex 3185	. . . 4 ⊢ (𝑉 ∈ 𝑋 → 𝑉 ∈ V)
3	2	adantr 480	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝑉 ∈ V)
4		elex 3185	. . . . 5 ⊢ (𝐸 ∈ 𝑌 → 𝐸 ∈ V)
5	4	adantl 481	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐸 ∈ V)
6	5	adantr 480	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝑣 = 𝑉) → 𝐸 ∈ V)
7		vex 3176	. . . 4 ⊢ 𝑣 ∈ V
8		rabexg 4739	. . . 4 ⊢ (𝑣 ∈ V → {𝑛 ∈ 𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒} ∈ V)
9	7, 8	mp1i 13	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → {𝑛 ∈ 𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒} ∈ V)
10		simprl 790	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → 𝑣 = 𝑉)
11		difeq1 3683	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (𝑣 ∖ {𝑛}) = (𝑉 ∖ {𝑛}))
12	11	adantr 480	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣 ∖ {𝑛}) = (𝑉 ∖ {𝑛}))
13		rneq 5272	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → ran 𝑒 = ran 𝐸)
14	13	eleq2d 2673	. . . . . . 7 ⊢ (𝑒 = 𝐸 → ({𝑘, 𝑛} ∈ ran 𝑒 ↔ {𝑘, 𝑛} ∈ ran 𝐸))
15	14	adantl 481	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ({𝑘, 𝑛} ∈ ran 𝑒 ↔ {𝑘, 𝑛} ∈ ran 𝐸))
16	12, 15	raleqbidv 3129	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸))
17	16	adantl 481	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸))
18	10, 17	rabeqbidv 3168	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → {𝑛 ∈ 𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒} = {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸})
19	3, 6, 9, 18	ovmpt2dv2 6692	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ( UnivVertex = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑛 ∈ 𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒}) → (𝑉 UnivVertex 𝐸) = {𝑛 ∈ 𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸}))
20	1, 19	mpi 20	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (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  Vcvv 3173   ∖ cdif 3537  {csn 4125  {cpr 4127  ran crn 5039  (class class class)co 6549   ↦ cmpt2 6551   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:  uvtxel  26017  uvtxisvtx  26018  uvtx0  26019  uvtx01vtx  26020  cusgrauvtxb  26024