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Theorem frgrawopreglem1 26337
Description: Lemma 1 for frgrawopreg 26342. 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 26285 . 2 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
2 usgrav 25633 . 2 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3 frgrawopreg.a . . . . 5 𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}
4 rabexg 4734 . . . . 5 (𝑉 ∈ V → {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾} ∈ V)
53, 4syl5eqel 2691 . . . 4 (𝑉 ∈ V → 𝐴 ∈ V)
6 frgrawopreg.b . . . . 5 𝐵 = (𝑉𝐴)
7 difexg 4730 . . . . 5 (𝑉 ∈ V → (𝑉𝐴) ∈ V)
86, 7syl5eqel 2691 . . . 4 (𝑉 ∈ V → 𝐵 ∈ V)
95, 8jca 552 . . 3 (𝑉 ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
109adantr 479 . 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 382   = wceq 1474  wcel 1976  {crab 2899  Vcvv 3172  cdif 3536   class class class wbr 4577  cfv 5790  (class class class)co 6527   USGrph cusg 25625   VDeg cvdg 26186   FriendGrph cfrgra 26281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
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 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5034  df-rel 5035  df-cnv 5036  df-dm 5038  df-rn 5039  df-usgra 25628  df-frgra 26282
This theorem is referenced by:  frgrawopreglem5  26341  frgrawopreg  26342  frgrawopreg1  26343  frgrawopreg2  26344
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