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Theorem isuvtx 25754
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 25689 . 2 UnivVertex = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑛𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒})
2 elex 3089 . . . 4 (𝑉𝑋𝑉 ∈ V)
32adantr 479 . . 3 ((𝑉𝑋𝐸𝑌) → 𝑉 ∈ V)
4 elex 3089 . . . . 5 (𝐸𝑌𝐸 ∈ V)
54adantl 480 . . . 4 ((𝑉𝑋𝐸𝑌) → 𝐸 ∈ V)
65adantr 479 . . 3 (((𝑉𝑋𝐸𝑌) ∧ 𝑣 = 𝑉) → 𝐸 ∈ V)
7 vex 3080 . . . 4 𝑣 ∈ V
8 rabexg 4638 . . . 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 789 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → 𝑣 = 𝑉)
11 difeq1 3587 . . . . . . 7 (𝑣 = 𝑉 → (𝑣 ∖ {𝑛}) = (𝑉 ∖ {𝑛}))
1211adantr 479 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 ∖ {𝑛}) = (𝑉 ∖ {𝑛}))
13 rneq 5163 . . . . . . . 8 (𝑒 = 𝐸 → ran 𝑒 = ran 𝐸)
1413eleq2d 2577 . . . . . . 7 (𝑒 = 𝐸 → ({𝑘, 𝑛} ∈ ran 𝑒 ↔ {𝑘, 𝑛} ∈ ran 𝐸))
1514adantl 480 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → ({𝑘, 𝑛} ∈ ran 𝑒 ↔ {𝑘, 𝑛} ∈ ran 𝐸))
1612, 15raleqbidv 3033 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸))
1716adantl 480 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸))
1810, 17rabeqbidv 3072 . . 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 6569 . 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 194  wa 382   = wceq 1474  wcel 1938  wral 2800  {crab 2804  Vcvv 3077  cdif 3441  {csn 4028  {cpr 4030  ran crn 4933  (class class class)co 6426  cmpt2 6428   UnivVertex cuvtx 25686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-iota 5653  df-fun 5691  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-uvtx 25689
This theorem is referenced by:  uvtxel  25755  uvtxisvtx  25756  uvtx0  25757  uvtx01vtx  25758  cusgrauvtxb  25762
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