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Theorem frgrwopreglem5lem 27472
 Description: Lemma for frgrwopreglem5 27473. (Contributed by AV, 5-Feb-2022.)
Hypotheses
Ref Expression
frgrwopreg.v 𝑉 = (Vtx‘𝐺)
frgrwopreg.d 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
frgrwopreg.b 𝐵 = (𝑉𝐴)
frgrwopreg.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frgrwopreglem5lem (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → ((𝐷𝑎) = (𝐷𝑥) ∧ (𝐷𝑎) ≠ (𝐷𝑏) ∧ (𝐷𝑥) ≠ (𝐷𝑦)))
Distinct variable groups:   𝑥,𝑉   𝑥,𝐴   𝑥,𝐺   𝑥,𝐾   𝑥,𝐷   𝐴,𝑏   𝑥,𝐵   𝑦,𝐷   𝐺,𝑎,𝑏,𝑦,𝑥   𝑦,𝑉
Allowed substitution hints:   𝐴(𝑦,𝑎)   𝐵(𝑦,𝑎,𝑏)   𝐷(𝑎,𝑏)   𝐸(𝑥,𝑦,𝑎,𝑏)   𝐾(𝑦,𝑎,𝑏)   𝑉(𝑎,𝑏)

Proof of Theorem frgrwopreglem5lem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 frgrwopreg.a . . . . . 6 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
21rabeq2i 3335 . . . . 5 (𝑥𝐴 ↔ (𝑥𝑉 ∧ (𝐷𝑥) = 𝐾))
3 fveq2 6350 . . . . . . . 8 (𝑥 = 𝑎 → (𝐷𝑥) = (𝐷𝑎))
43eqeq1d 2760 . . . . . . 7 (𝑥 = 𝑎 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑎) = 𝐾))
54, 1elrab2 3505 . . . . . 6 (𝑎𝐴 ↔ (𝑎𝑉 ∧ (𝐷𝑎) = 𝐾))
6 eqtr3 2779 . . . . . . . . 9 (((𝐷𝑎) = 𝐾 ∧ (𝐷𝑥) = 𝐾) → (𝐷𝑎) = (𝐷𝑥))
76expcom 450 . . . . . . . 8 ((𝐷𝑥) = 𝐾 → ((𝐷𝑎) = 𝐾 → (𝐷𝑎) = (𝐷𝑥)))
87adantl 473 . . . . . . 7 ((𝑥𝑉 ∧ (𝐷𝑥) = 𝐾) → ((𝐷𝑎) = 𝐾 → (𝐷𝑎) = (𝐷𝑥)))
98com12 32 . . . . . 6 ((𝐷𝑎) = 𝐾 → ((𝑥𝑉 ∧ (𝐷𝑥) = 𝐾) → (𝐷𝑎) = (𝐷𝑥)))
105, 9simplbiim 661 . . . . 5 (𝑎𝐴 → ((𝑥𝑉 ∧ (𝐷𝑥) = 𝐾) → (𝐷𝑎) = (𝐷𝑥)))
112, 10syl5bi 232 . . . 4 (𝑎𝐴 → (𝑥𝐴 → (𝐷𝑎) = (𝐷𝑥)))
1211imp 444 . . 3 ((𝑎𝐴𝑥𝐴) → (𝐷𝑎) = (𝐷𝑥))
1312adantr 472 . 2 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → (𝐷𝑎) = (𝐷𝑥))
14 frgrwopreg.v . . . 4 𝑉 = (Vtx‘𝐺)
15 frgrwopreg.d . . . 4 𝐷 = (VtxDeg‘𝐺)
16 frgrwopreg.b . . . 4 𝐵 = (𝑉𝐴)
1714, 15, 1, 16frgrwopreglem3 27466 . . 3 ((𝑎𝐴𝑏𝐵) → (𝐷𝑎) ≠ (𝐷𝑏))
1817ad2ant2r 800 . 2 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → (𝐷𝑎) ≠ (𝐷𝑏))
19 fveq2 6350 . . . . . . 7 (𝑥 = 𝑧 → (𝐷𝑥) = (𝐷𝑧))
2019eqeq1d 2760 . . . . . 6 (𝑥 = 𝑧 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑧) = 𝐾))
2120cbvrabv 3337 . . . . 5 {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} = {𝑧𝑉 ∣ (𝐷𝑧) = 𝐾}
221, 21eqtri 2780 . . . 4 𝐴 = {𝑧𝑉 ∣ (𝐷𝑧) = 𝐾}
2314, 15, 22, 16frgrwopreglem3 27466 . . 3 ((𝑥𝐴𝑦𝐵) → (𝐷𝑥) ≠ (𝐷𝑦))
2423ad2ant2l 799 . 2 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → (𝐷𝑥) ≠ (𝐷𝑦))
2513, 18, 243jca 1123 1 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → ((𝐷𝑎) = (𝐷𝑥) ∧ (𝐷𝑎) ≠ (𝐷𝑏) ∧ (𝐷𝑥) ≠ (𝐷𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1630   ∈ wcel 2137   ≠ wne 2930  {crab 3052   ∖ cdif 3710  ‘cfv 6047  Vtxcvtx 26071  Edgcedg 26136  VtxDegcvtxdg 26569 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-iota 6010  df-fv 6055 This theorem is referenced by:  frgrwopreglem5  27473
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