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Theorem gaorb 18198
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 6978 . . . . . 6 (𝑥 = 𝐴 → (𝑔 𝑥) = (𝑔 𝐴))
2 eqeq12 2785 . . . . . 6 (((𝑔 𝑥) = (𝑔 𝐴) ∧ 𝑦 = 𝐵) → ((𝑔 𝑥) = 𝑦 ↔ (𝑔 𝐴) = 𝐵))
31, 2sylan 572 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑔 𝑥) = 𝑦 ↔ (𝑔 𝐴) = 𝐵))
43rexbidv 3236 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑔𝑋 (𝑔 𝑥) = 𝑦 ↔ ∃𝑔𝑋 (𝑔 𝐴) = 𝐵))
5 oveq1 6977 . . . . . 6 (𝑔 = → (𝑔 𝐴) = ( 𝐴))
65eqeq1d 2774 . . . . 5 (𝑔 = → ((𝑔 𝐴) = 𝐵 ↔ ( 𝐴) = 𝐵))
76cbvrexv 3378 . . . 4 (∃𝑔𝑋 (𝑔 𝐴) = 𝐵 ↔ ∃𝑋 ( 𝐴) = 𝐵)
84, 7syl6bb 279 . . 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 3412 . . . . . . 7 𝑥 ∈ V
11 vex 3412 . . . . . . 7 𝑦 ∈ V
1210, 11prss 4621 . . . . . 6 ((𝑥𝑌𝑦𝑌) ↔ {𝑥, 𝑦} ⊆ 𝑌)
1312anbi1i 614 . . . . 5 (((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦))
1413opabbii 4990 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
159, 14eqtr4i 2799 . . 3 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
168, 15brab2a 5487 . 2 (𝐴 𝐵 ↔ ((𝐴𝑌𝐵𝑌) ∧ ∃𝑋 ( 𝐴) = 𝐵))
17 df-3an 1070 . 2 ((𝐴𝑌𝐵𝑌 ∧ ∃𝑋 ( 𝐴) = 𝐵) ↔ ((𝐴𝑌𝐵𝑌) ∧ ∃𝑋 ( 𝐴) = 𝐵))
1816, 17bitr4i 270 1 (𝐴 𝐵 ↔ (𝐴𝑌𝐵𝑌 ∧ ∃𝑋 ( 𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2048  wrex 3083  wss 3825  {cpr 4437   class class class wbr 4923  {copab 4985  (class class class)co 6970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-xp 5406  df-iota 6146  df-fv 6190  df-ov 6973
This theorem is referenced by:  gaorber  18199  orbsta  18204  sylow2alem1  18493  sylow2alem2  18494  sylow3lem3  18505
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