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Theorem frgrwopreglem5lem 27474
Description: Lemma for frgrwopreglem5 27475. (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 3337 . . . . 5 (𝑥𝐴 ↔ (𝑥𝑉 ∧ (𝐷𝑥) = 𝐾))
3 fveq2 6352 . . . . . . . 8 (𝑥 = 𝑎 → (𝐷𝑥) = (𝐷𝑎))
43eqeq1d 2762 . . . . . . 7 (𝑥 = 𝑎 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑎) = 𝐾))
54, 1elrab2 3507 . . . . . 6 (𝑎𝐴 ↔ (𝑎𝑉 ∧ (𝐷𝑎) = 𝐾))
6 eqtr3 2781 . . . . . . . . 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 27468 . . 3 ((𝑎𝐴𝑏𝐵) → (𝐷𝑎) ≠ (𝐷𝑏))
1817ad2ant2r 800 . 2 (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → (𝐷𝑎) ≠ (𝐷𝑏))
19 fveq2 6352 . . . . . . 7 (𝑥 = 𝑧 → (𝐷𝑥) = (𝐷𝑧))
2019eqeq1d 2762 . . . . . 6 (𝑥 = 𝑧 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑧) = 𝐾))
2120cbvrabv 3339 . . . . 5 {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} = {𝑧𝑉 ∣ (𝐷𝑧) = 𝐾}
221, 21eqtri 2782 . . . 4 𝐴 = {𝑧𝑉 ∣ (𝐷𝑧) = 𝐾}
2314, 15, 22, 16frgrwopreglem3 27468 . . 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 1632  wcel 2139  wne 2932  {crab 3054  cdif 3712  cfv 6049  Vtxcvtx 26073  Edgcedg 26138  VtxDegcvtxdg 26571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057
This theorem is referenced by:  frgrwopreglem5  27475
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