Theorem gaorb 18828

Description: The orbit equivalence relation puts two points in the group action in the same equivalence class iff there is a group element that takes one element to the other. (Contributed by Mario Carneiro, 14-Jan-2015.)

Hypothesis

Ref	Expression
gaorb.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}

Assertion

Ref	Expression
gaorb	⊢ (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵))

Distinct variable groups:   𝑔,ℎ,𝑥,𝑦,𝐴   𝐵,𝑔,ℎ,𝑥,𝑦   ∼ ,ℎ   ⊕ ,𝑔,ℎ,𝑥,𝑦   𝑔,𝑋,ℎ,𝑥,𝑦   ℎ,𝑌,𝑥,𝑦

Allowed substitution hints:   ∼ (𝑥,𝑦,𝑔)   𝑌(𝑔)

Proof of Theorem gaorb

Step	Hyp	Ref	Expression
1		oveq2 7263	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑔 ⊕ 𝑥) = (𝑔 ⊕ 𝐴))
2		eqeq12 2755	. . . . . 6 ⊢ (((𝑔 ⊕ 𝑥) = (𝑔 ⊕ 𝐴) ∧ 𝑦 = 𝐵) → ((𝑔 ⊕ 𝑥) = 𝑦 ↔ (𝑔 ⊕ 𝐴) = 𝐵))
3	1, 2	sylan 579	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑔 ⊕ 𝑥) = 𝑦 ↔ (𝑔 ⊕ 𝐴) = 𝐵))
4	3	rexbidv 3225	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦 ↔ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝐴) = 𝐵))
5		oveq1 7262	. . . . . 6 ⊢ (𝑔 = ℎ → (𝑔 ⊕ 𝐴) = (ℎ ⊕ 𝐴))
6	5	eqeq1d 2740	. . . . 5 ⊢ (𝑔 = ℎ → ((𝑔 ⊕ 𝐴) = 𝐵 ↔ (ℎ ⊕ 𝐴) = 𝐵))
7	6	cbvrexvw 3373	. . . 4 ⊢ (∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝐴) = 𝐵 ↔ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵)
8	4, 7	bitrdi 286	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦 ↔ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵))
9		gaorb.1	. . . 4 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
10		vex 3426	. . . . . . 7 ⊢ 𝑥 ∈ V
11		vex 3426	. . . . . . 7 ⊢ 𝑦 ∈ V
12	10, 11	prss 4750	. . . . . 6 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ↔ {𝑥, 𝑦} ⊆ 𝑌)
13	12	anbi1i 623	. . . . 5 ⊢ (((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦))
14	13	opabbii 5137	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
15	9, 14	eqtr4i 2769	. . 3 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
16	8, 15	brab2a 5670	. 2 ⊢ (𝐴 ∼ 𝐵 ↔ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵))
17		df-3an 1087	. 2 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵) ↔ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵))
18	16, 17	bitr4i 277	1 ⊢ (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   ⊆ wss 3883  {cpr 4560   class class class wbr 5070  {copab 5132  (class class class)co 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-iota 6376  df-fv 6426  df-ov 7258

This theorem is referenced by:  gaorber  18829  orbsta  18834  sylow2alem1  19137  sylow2alem2  19138  sylow3lem3  19149  lsmsnorb  31481