Theorem frgrwopreglem5lem 28101

Description: Lemma for frgrwopreglem5 28102. (Contributed by AV, 5-Feb-2022.)

Hypotheses

Ref	Expression
frgrwopreg.v	⊢ 𝑉 = (Vtx‘𝐺)
frgrwopreg.d	⊢ 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a	⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
frgrwopreg.b	⊢ 𝐵 = (𝑉 ∖ 𝐴)
frgrwopreg.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
frgrwopreglem5lem	⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐷‘𝑎) = (𝐷‘𝑥) ∧ (𝐷‘𝑎) ≠ (𝐷‘𝑏) ∧ (𝐷‘𝑥) ≠ (𝐷‘𝑦)))

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.

Step	Hyp	Ref	Expression
1		frgrwopreg.a	. . . . . 6 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
2	1	rabeq2i 3489	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾))
3		fveqeq2 6681	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑎) = 𝐾))
4	3, 1	elrab2 3685	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 ↔ (𝑎 ∈ 𝑉 ∧ (𝐷‘𝑎) = 𝐾))
5		eqtr3 2845	. . . . . . . . 9 ⊢ (((𝐷‘𝑎) = 𝐾 ∧ (𝐷‘𝑥) = 𝐾) → (𝐷‘𝑎) = (𝐷‘𝑥))
6	5	expcom 416	. . . . . . . 8 ⊢ ((𝐷‘𝑥) = 𝐾 → ((𝐷‘𝑎) = 𝐾 → (𝐷‘𝑎) = (𝐷‘𝑥)))
7	6	adantl 484	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) → ((𝐷‘𝑎) = 𝐾 → (𝐷‘𝑎) = (𝐷‘𝑥)))
8	7	com12 32	. . . . . 6 ⊢ ((𝐷‘𝑎) = 𝐾 → ((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) → (𝐷‘𝑎) = (𝐷‘𝑥)))
9	4, 8	simplbiim 507	. . . . 5 ⊢ (𝑎 ∈ 𝐴 → ((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) → (𝐷‘𝑎) = (𝐷‘𝑥)))
10	2, 9	syl5bi 244	. . . 4 ⊢ (𝑎 ∈ 𝐴 → (𝑥 ∈ 𝐴 → (𝐷‘𝑎) = (𝐷‘𝑥)))
11	10	imp 409	. . 3 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐷‘𝑎) = (𝐷‘𝑥))
12	11	adantr 483	. 2 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐷‘𝑎) = (𝐷‘𝑥))
13		frgrwopreg.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
14		frgrwopreg.d	. . . 4 ⊢ 𝐷 = (VtxDeg‘𝐺)
15		frgrwopreg.b	. . . 4 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
16	13, 14, 1, 15	frgrwopreglem3 28095	. . 3 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐷‘𝑎) ≠ (𝐷‘𝑏))
17	16	ad2ant2r 745	. 2 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐷‘𝑎) ≠ (𝐷‘𝑏))
18		fveqeq2 6681	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑧) = 𝐾))
19	18	cbvrabv 3493	. . . . 5 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = {𝑧 ∈ 𝑉 ∣ (𝐷‘𝑧) = 𝐾}
20	1, 19	eqtri 2846	. . . 4 ⊢ 𝐴 = {𝑧 ∈ 𝑉 ∣ (𝐷‘𝑧) = 𝐾}
21	13, 14, 20, 15	frgrwopreglem3 28095	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐷‘𝑥) ≠ (𝐷‘𝑦))
22	21	ad2ant2l 744	. 2 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐷‘𝑥) ≠ (𝐷‘𝑦))
23	12, 17, 22	3jca 1124	1 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐷‘𝑎) = (𝐷‘𝑥) ∧ (𝐷‘𝑎) ≠ (𝐷‘𝑏) ∧ (𝐷‘𝑥) ≠ (𝐷‘𝑦)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  {crab 3144   ∖ cdif 3935  ‘cfv 6357  Vtxcvtx 26783  Edgcedg 26834  VtxDegcvtxdg 27249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365

This theorem is referenced by:  frgrwopreglem5  28102