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Theorem gaorb 17656
 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 6613 . . . . . 6 (𝑥 = 𝐴 → (𝑔 𝑥) = (𝑔 𝐴))
2 eqeq12 2639 . . . . . 6 (((𝑔 𝑥) = (𝑔 𝐴) ∧ 𝑦 = 𝐵) → ((𝑔 𝑥) = 𝑦 ↔ (𝑔 𝐴) = 𝐵))
31, 2sylan 488 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑔 𝑥) = 𝑦 ↔ (𝑔 𝐴) = 𝐵))
43rexbidv 3050 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑔𝑋 (𝑔 𝑥) = 𝑦 ↔ ∃𝑔𝑋 (𝑔 𝐴) = 𝐵))
5 oveq1 6612 . . . . . 6 (𝑔 = → (𝑔 𝐴) = ( 𝐴))
65eqeq1d 2628 . . . . 5 (𝑔 = → ((𝑔 𝐴) = 𝐵 ↔ ( 𝐴) = 𝐵))
76cbvrexv 3165 . . . 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 3194 . . . . . . 7 𝑥 ∈ V
11 vex 3194 . . . . . . 7 𝑦 ∈ V
1210, 11prss 4324 . . . . . 6 ((𝑥𝑌𝑦𝑌) ↔ {𝑥, 𝑦} ⊆ 𝑌)
1312anbi1i 730 . . . . 5 (((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦))
1413opabbii 4684 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
159, 14eqtr4i 2651 . . 3 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑌𝑦𝑌) ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
168, 15brab2ga 5160 . 2 (𝐴 𝐵 ↔ ((𝐴𝑌𝐵𝑌) ∧ ∃𝑋 ( 𝐴) = 𝐵))
17 df-3an 1038 . 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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1992  ∃wrex 2913   ⊆ wss 3560  {cpr 4155   class class class wbr 4618  {copab 4677  (class class class)co 6605 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-iota 5813  df-fv 5858  df-ov 6608 This theorem is referenced by:  gaorber  17657  orbsta  17662  sylow2alem1  17948  sylow2alem2  17949  sylow3lem3  17960
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