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Theorem frgrwopreglem1 27292
Description: Lemma 1 for frgrwopreg 27303: 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.) (Revised by AV, 10-May-2021.)
Hypotheses
Ref Expression
frgrwopreg.v 𝑉 = (Vtx‘𝐺)
frgrwopreg.d 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
frgrwopreg.b 𝐵 = (𝑉𝐴)
Assertion
Ref Expression
frgrwopreglem1 (𝐴 ∈ V ∧ 𝐵 ∈ V)
Distinct variable group:   𝑥,𝑉
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐷(𝑥)   𝐺(𝑥)   𝐾(𝑥)

Proof of Theorem frgrwopreglem1
StepHypRef Expression
1 frgrwopreg.v . . 3 𝑉 = (Vtx‘𝐺)
2 fvex 6239 . . 3 (Vtx‘𝐺) ∈ V
31, 2eqeltri 2726 . 2 𝑉 ∈ V
4 frgrwopreg.a . . . 4 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
5 rabexg 4844 . . . 4 (𝑉 ∈ V → {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} ∈ V)
64, 5syl5eqel 2734 . . 3 (𝑉 ∈ V → 𝐴 ∈ V)
7 frgrwopreg.b . . . 4 𝐵 = (𝑉𝐴)
8 difexg 4841 . . . 4 (𝑉 ∈ V → (𝑉𝐴) ∈ V)
97, 8syl5eqel 2734 . . 3 (𝑉 ∈ V → 𝐵 ∈ V)
106, 9jca 553 . 2 (𝑉 ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
113, 10ax-mp 5 1 (𝐴 ∈ V ∧ 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  cdif 3604  cfv 5926  Vtxcvtx 25919  VtxDegcvtxdg 26417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213  df-uni 4469  df-iota 5889  df-fv 5934
This theorem is referenced by:  frgrwopreg2  27299  frgrwopreglem5  27301  frgrwopreglem5ALT  27302  frgrwopreg  27303
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