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Theorem gaorb 18901
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 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
Assertion
Ref Expression
gaorb (𝐴 𝐵 ↔ (𝐴𝑌𝐵𝑌 ∧ ∃𝑋 ( 𝐴) = 𝐵))
Distinct variable groups:   𝑔,,𝑥,𝑦,𝐴   𝐵,𝑔,,𝑥,𝑦   ,   ,𝑔,,𝑥,𝑦   𝑔,𝑋,,𝑥,𝑦   ,𝑌,𝑥,𝑦
Allowed substitution hints:   (𝑥,𝑦,𝑔)   𝑌(𝑔)

Proof of Theorem gaorb
StepHypRef Expression
1 oveq2 7276 . . . . . 6 (𝑥 = 𝐴 → (𝑔 𝑥) = (𝑔 𝐴))
2 eqeq12 2755 . . . . . 6 (((𝑔 𝑥) = (𝑔 𝐴) ∧ 𝑦 = 𝐵) → ((𝑔 𝑥) = 𝑦 ↔ (𝑔 𝐴) = 𝐵))
31, 2sylan 580 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑔 𝑥) = 𝑦 ↔ (𝑔 𝐴) = 𝐵))
43rexbidv 3224 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑔𝑋 (𝑔 𝑥) = 𝑦 ↔ ∃𝑔𝑋 (𝑔 𝐴) = 𝐵))
5 oveq1 7275 . . . . . 6 (𝑔 = → (𝑔 𝐴) = ( 𝐴))
65eqeq1d 2740 . . . . 5 (𝑔 = → ((𝑔 𝐴) = 𝐵 ↔ ( 𝐴) = 𝐵))
76cbvrexvw 3382 . . . 4 (∃𝑔𝑋 (𝑔 𝐴) = 𝐵 ↔ ∃𝑋 ( 𝐴) = 𝐵)
84, 7bitrdi 287 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑔𝑋 (𝑔 𝑥) = 𝑦 ↔ ∃𝑋 ( 𝐴) = 𝐵))
9 gaorb.1 . . . 4 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
10 vex 3434 . . . . . . 7 𝑥 ∈ V
11 vex 3434 . . . . . . 7 𝑦 ∈ V
1210, 11prss 4754 . . . . . 6 ((𝑥𝑌𝑦𝑌) ↔ {𝑥, 𝑦} ⊆ 𝑌)
1312anbi1i 624 . . . . 5 (((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦))
1413opabbii 5141 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
159, 14eqtr4i 2769 . . 3 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
168, 15brab2a 5675 . 2 (𝐴 𝐵 ↔ ((𝐴𝑌𝐵𝑌) ∧ ∃𝑋 ( 𝐴) = 𝐵))
17 df-3an 1088 . 2 ((𝐴𝑌𝐵𝑌 ∧ ∃𝑋 ( 𝐴) = 𝐵) ↔ ((𝐴𝑌𝐵𝑌) ∧ ∃𝑋 ( 𝐴) = 𝐵))
1816, 17bitr4i 277 1 (𝐴 𝐵 ↔ (𝐴𝑌𝐵𝑌 ∧ ∃𝑋 ( 𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wrex 3065  wss 3887  {cpr 4564   class class class wbr 5074  {copab 5136  (class class class)co 7268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-opab 5137  df-xp 5591  df-iota 6385  df-fv 6435  df-ov 7271
This theorem is referenced by:  gaorber  18902  orbsta  18907  sylow2alem1  19210  sylow2alem2  19211  sylow3lem3  19222  lsmsnorb  31565
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