Theorem gaorb 19423

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 7427	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑔 ⊕ 𝑥) = (𝑔 ⊕ 𝐴))
2		eqeq12 2782	. . . . . 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 3191	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦 ↔ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝐴) = 𝐵))
5		oveq1 7426	. . . . . 6 ⊢ (𝑔 = ℎ → (𝑔 ⊕ 𝐴) = (ℎ ⊕ 𝐴))
6	5	eqeq1d 2767	. . . . 5 ⊢ (𝑔 = ℎ → ((𝑔 ⊕ 𝐴) = 𝐵 ↔ (ℎ ⊕ 𝐴) = 𝐵))
7	6	cbvrexvw 3246	. . . 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 3461	. . . . . . 7 ⊢ 𝑥 ∈ V
11		vex 3461	. . . . . . 7 ⊢ 𝑦 ∈ V
12	10, 11	prss 4788	. . . . . 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 5180	. . . 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 2791	. . 3 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
16	8, 15	brab2a 5756	. 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 2146  ∃wrex 3091   ⊆ wss 3906  {cpr 4593   class class class wbr 5111  {copab 5175  (class class class)co 7419

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-iota 6496  df-fv 6548  df-ov 7422

This theorem is used by:  gaorber  19424  orbsta  19429  sylow2alem1  19733  sylow2alem2  19734  sylow3lem3  19745  lsmsnorb  33770