MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gaorb Structured version   Visualization version   GIF version

Theorem gaorb 18375
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 7153 . . . . . 6 (𝑥 = 𝐴 → (𝑔 𝑥) = (𝑔 𝐴))
2 eqeq12 2832 . . . . . 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 3294 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑔𝑋 (𝑔 𝑥) = 𝑦 ↔ ∃𝑔𝑋 (𝑔 𝐴) = 𝐵))
5 oveq1 7152 . . . . . 6 (𝑔 = → (𝑔 𝐴) = ( 𝐴))
65eqeq1d 2820 . . . . 5 (𝑔 = → ((𝑔 𝐴) = 𝐵 ↔ ( 𝐴) = 𝐵))
76cbvrexvw 3448 . . . 4 (∃𝑔𝑋 (𝑔 𝐴) = 𝐵 ↔ ∃𝑋 ( 𝐴) = 𝐵)
84, 7syl6bb 288 . . 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 3495 . . . . . . 7 𝑥 ∈ V
11 vex 3495 . . . . . . 7 𝑦 ∈ V
1210, 11prss 4745 . . . . . 6 ((𝑥𝑌𝑦𝑌) ↔ {𝑥, 𝑦} ⊆ 𝑌)
1312anbi1i 623 . . . . 5 (((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦))
1413opabbii 5124 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
159, 14eqtr4i 2844 . . 3 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
168, 15brab2a 5637 . 2 (𝐴 𝐵 ↔ ((𝐴𝑌𝐵𝑌) ∧ ∃𝑋 ( 𝐴) = 𝐵))
17 df-3an 1081 . 2 ((𝐴𝑌𝐵𝑌 ∧ ∃𝑋 ( 𝐴) = 𝐵) ↔ ((𝐴𝑌𝐵𝑌) ∧ ∃𝑋 ( 𝐴) = 𝐵))
1816, 17bitr4i 279 1 (𝐴 𝐵 ↔ (𝐴𝑌𝐵𝑌 ∧ ∃𝑋 ( 𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  wss 3933  {cpr 4559   class class class wbr 5057  {copab 5119  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by:  gaorber  18376  orbsta  18381  sylow2alem1  18671  sylow2alem2  18672  sylow3lem3  18683
  Copyright terms: Public domain W3C validator