Theorem gaorb 19572

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 7459	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑔 ⊕ 𝑥) = (𝑔 ⊕ 𝐴))
2		eqeq12 2780	. . . . . 6 ⊢ (((𝑔 ⊕ 𝑥) = (𝑔 ⊕ 𝐴) ∧ 𝑦 = 𝐵) → ((𝑔 ⊕ 𝑥) = 𝑦 ↔ (𝑔 ⊕ 𝐴) = 𝐵))
3	1, 2	sylan 592	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑔 ⊕ 𝑥) = 𝑦 ↔ (𝑔 ⊕ 𝐴) = 𝐵))
4	3	rexbidv 3190	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦 ↔ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝐴) = 𝐵))
5		oveq1 7458	. . . . . 6 ⊢ (𝑔 = ℎ → (𝑔 ⊕ 𝐴) = (ℎ ⊕ 𝐴))
6	5	eqeq1d 2765	. . . . 5 ⊢ (𝑔 = ℎ → ((𝑔 ⊕ 𝐴) = 𝐵 ↔ (ℎ ⊕ 𝐴) = 𝐵))
7	6	cbvrexvw 3245	. . . 4 ⊢ (∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝐴) = 𝐵 ↔ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵)
8	4, 7	bitrdi 290	. . 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 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
11		vex 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
12	10, 11	prss 4781	. . . . . 6 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ↔ {𝑥, 𝑦} ⊆ 𝑌)
13	12	anbi1i 636	. . . . 5 ⊢ (((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦))
14	13	opabbii 5178	. . . 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 2789	. . 3 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
16	8, 15	brab2a 5752	. 2 ⊢ (𝐴 ∼ 𝐵 ↔ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵))
17		df-3an 1105	. 2 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵) ↔ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵))
18	16, 17	bitr4i 281	1 ⊢ (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  {cpr 4590   class class class wbr 5108  {copab 5173  (class class class)co 7451

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-xp 5664  df-iota 6520  df-fv 6574  df-ov 7454

This theorem is used by:  gaorber  19573  orbsta  19578  sylow2alem1  19882  sylow2alem2  19883  sylow3lem3  19894  lsmsnorb  34050