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Theorem gaorb 15086
 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
Assertion
Ref Expression
gaorb
Distinct variable groups:   ,,,,   ,,,,   ,   ,,,,   ,,,,   ,,,
Allowed substitution hints:   (,,)   ()

Proof of Theorem gaorb
StepHypRef Expression
1 oveq2 6091 . . . . . 6
2 eqeq12 2450 . . . . . 6
31, 2sylan 459 . . . . 5
43rexbidv 2728 . . . 4
5 oveq1 6090 . . . . . 6
65eqeq1d 2446 . . . . 5
76cbvrexv 2935 . . . 4
84, 7syl6bb 254 . . 3
9 gaorb.1 . . . 4
10 vex 2961 . . . . . . 7
11 vex 2961 . . . . . . 7
1210, 11prss 3954 . . . . . 6
1312anbi1i 678 . . . . 5
1413opabbii 4274 . . . 4
159, 14eqtr4i 2461 . . 3
168, 15brab2ga 4953 . 2
17 df-3an 939 . 2
1816, 17bitr4i 245 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726  wrex 2708   wss 3322  cpr 3817   class class class wbr 4214  copab 4267  (class class class)co 6083 This theorem is referenced by:  gaorber  15087  orbsta  15092  sylow2alem1  15253  sylow2alem2  15254  sylow3lem3  15265 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-iota 5420  df-fv 5464  df-ov 6086
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