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Theorem isuvtx 26016
 Description: The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
isuvtx ((𝑉𝑋𝐸𝑌) → (𝑉 UnivVertex 𝐸) = {𝑛𝑉 ∣ ∀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.
StepHypRef Expression
1 df-uvtx 25951 . 2 UnivVertex = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑛𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒})
2 elex 3185 . . . 4 (𝑉𝑋𝑉 ∈ V)
32adantr 480 . . 3 ((𝑉𝑋𝐸𝑌) → 𝑉 ∈ V)
4 elex 3185 . . . . 5 (𝐸𝑌𝐸 ∈ V)
54adantl 481 . . . 4 ((𝑉𝑋𝐸𝑌) → 𝐸 ∈ V)
65adantr 480 . . 3 (((𝑉𝑋𝐸𝑌) ∧ 𝑣 = 𝑉) → 𝐸 ∈ V)
7 vex 3176 . . . 4 𝑣 ∈ V
8 rabexg 4739 . . . 4 (𝑣 ∈ V → {𝑛𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒} ∈ V)
97, 8mp1i 13 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → {𝑛𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒} ∈ V)
10 simprl 790 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → 𝑣 = 𝑉)
11 difeq1 3683 . . . . . . 7 (𝑣 = 𝑉 → (𝑣 ∖ {𝑛}) = (𝑉 ∖ {𝑛}))
1211adantr 480 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 ∖ {𝑛}) = (𝑉 ∖ {𝑛}))
13 rneq 5272 . . . . . . . 8 (𝑒 = 𝐸 → ran 𝑒 = ran 𝐸)
1413eleq2d 2673 . . . . . . 7 (𝑒 = 𝐸 → ({𝑘, 𝑛} ∈ ran 𝑒 ↔ {𝑘, 𝑛} ∈ ran 𝐸))
1514adantl 481 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → ({𝑘, 𝑛} ∈ ran 𝑒 ↔ {𝑘, 𝑛} ∈ ran 𝐸))
1612, 15raleqbidv 3129 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸))
1716adantl 481 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸))
1810, 17rabeqbidv 3168 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → {𝑛𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒} = {𝑛𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸})
193, 6, 9, 18ovmpt2dv2 6692 . 2 ((𝑉𝑋𝐸𝑌) → ( UnivVertex = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑛𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒}) → (𝑉 UnivVertex 𝐸) = {𝑛𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸}))
201, 19mpi 20 1 ((𝑉𝑋𝐸𝑌) → (𝑉 UnivVertex 𝐸) = {𝑛𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸})
 Colors of variables: wff setvar class 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
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