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Theorem frgrawopreglem1 26571
Description: Lemma 1 for frgrawopreg 26576. In a friendship graph, the classes A and B are sets. The definition of A and B corresponds to definition 3 in [Huneke] p. 2: "Let A be the set of all vertices of degree k, let B be the set of all vertices of degree different from k, ..." (Contributed by Alexander van der Vekens, 31-Dec-2017.)
Hypotheses
Ref Expression
frgrawopreg.a 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
frgrawopreg.b 𝐵 = (𝑉𝐴)
Assertion
Ref Expression
frgrawopreglem1 (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐸   𝑥,𝐾   𝑥,𝑉
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem frgrawopreglem1
StepHypRef Expression
1 frisusgra 26519 . 2 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
2 usgrav 25867 . 2 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3 frgrawopreg.a . . . . 5 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
4 rabexg 4739 . . . . 5 (𝑉 ∈ V → {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} ∈ V)
53, 4syl5eqel 2692 . . . 4 (𝑉 ∈ V → 𝐴 ∈ V)
6 frgrawopreg.b . . . . 5 𝐵 = (𝑉𝐴)
7 difexg 4735 . . . . 5 (𝑉 ∈ V → (𝑉𝐴) ∈ V)
86, 7syl5eqel 2692 . . . 4 (𝑉 ∈ V → 𝐵 ∈ V)
95, 8jca 553 . . 3 (𝑉 ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
109adantr 480 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
111, 2, 103syl 18 1 (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  {crab 2900  Vcvv 3173  cdif 3537   class class class wbr 4583  cfv 5804  (class class class)co 6549   USGrph cusg 25859   VDeg cvdg 26420   FriendGrph cfrgra 26515
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-reu 2903  df-rab 2905  df-v 3175  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-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-usgra 25862  df-frgra 26516
This theorem is referenced by:  frgrawopreglem5  26575  frgrawopreg  26576  frgrawopreg1  26577  frgrawopreg2  26578
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