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Theorem gaorb 17829
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 6741 . . . . . 6 (𝑥 = 𝐴 → (𝑔 𝑥) = (𝑔 𝐴))
2 eqeq12 2705 . . . . . 6 (((𝑔 𝑥) = (𝑔 𝐴) ∧ 𝑦 = 𝐵) → ((𝑔 𝑥) = 𝑦 ↔ (𝑔 𝐴) = 𝐵))
31, 2sylan 489 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑔 𝑥) = 𝑦 ↔ (𝑔 𝐴) = 𝐵))
43rexbidv 3122 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑔𝑋 (𝑔 𝑥) = 𝑦 ↔ ∃𝑔𝑋 (𝑔 𝐴) = 𝐵))
5 oveq1 6740 . . . . . 6 (𝑔 = → (𝑔 𝐴) = ( 𝐴))
65eqeq1d 2694 . . . . 5 (𝑔 = → ((𝑔 𝐴) = 𝐵 ↔ ( 𝐴) = 𝐵))
76cbvrexv 3243 . . . 4 (∃𝑔𝑋 (𝑔 𝐴) = 𝐵 ↔ ∃𝑋 ( 𝐴) = 𝐵)
84, 7syl6bb 276 . . 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 3275 . . . . . . 7 𝑥 ∈ V
11 vex 3275 . . . . . . 7 𝑦 ∈ V
1210, 11prss 4427 . . . . . 6 ((𝑥𝑌𝑦𝑌) ↔ {𝑥, 𝑦} ⊆ 𝑌)
1312anbi1i 733 . . . . 5 (((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦))
1413opabbii 4793 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
159, 14eqtr4i 2717 . . 3 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
168, 15brab2a 5271 . 2 (𝐴 𝐵 ↔ ((𝐴𝑌𝐵𝑌) ∧ ∃𝑋 ( 𝐴) = 𝐵))
17 df-3an 1074 . 2 ((𝐴𝑌𝐵𝑌 ∧ ∃𝑋 ( 𝐴) = 𝐵) ↔ ((𝐴𝑌𝐵𝑌) ∧ ∃𝑋 ( 𝐴) = 𝐵))
1816, 17bitr4i 267 1 (𝐴 𝐵 ↔ (𝐴𝑌𝐵𝑌 ∧ ∃𝑋 ( 𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1564  wcel 2071  wrex 2983  wss 3648  {cpr 4255   class class class wbr 4728  {copab 4788  (class class class)co 6733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pr 4979
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ral 2987  df-rex 2988  df-rab 2991  df-v 3274  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-br 4729  df-opab 4789  df-xp 5192  df-iota 5932  df-fv 5977  df-ov 6736
This theorem is referenced by:  gaorber  17830  orbsta  17835  sylow2alem1  18121  sylow2alem2  18122  sylow3lem3  18133
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