Theorem gaorb 17740

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 } C_ Y /\ E. g e. X ( g .(+) x ) = y ) }

Assertion

Ref	Expression
gaorb	|- ( A .~ B <-> ( A e. Y /\ B e. Y /\ E. h e. X ( h .(+) A ) = B ) )

Distinct variable groups:   g, h, x, y, A   B, g, h, x, y   .~ , h   .(+) , g, h, x, y   g, X, h, x, y   h, Y, x, y

Allowed substitution hints:   .~ ( x, y, g)   Y( g)

Proof of Theorem gaorb

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . 6 |- ( x = A -> ( g .(+) x ) = ( g .(+) A ) )
2		eqeq12 2635	. . . . . 6 |- ( ( ( g .(+) x ) = ( g .(+) A ) /\ y = B ) -> ( ( g .(+) x ) = y <-> ( g .(+) A ) = B ) )
3	1, 2	sylan 488	. . . . 5 |- ( ( x = A /\ y = B ) -> ( ( g .(+) x ) = y <-> ( g .(+) A ) = B ) )
4	3	rexbidv 3052	. . . 4 |- ( ( x = A /\ y = B ) -> ( E. g e. X ( g .(+) x ) = y <-> E. g e. X ( g .(+) A ) = B ) )
5		oveq1 6657	. . . . . 6 |- ( g = h -> ( g .(+) A ) = ( h .(+) A ) )
6	5	eqeq1d 2624	. . . . 5 |- ( g = h -> ( ( g .(+) A ) = B <-> ( h .(+) A ) = B ) )
7	6	cbvrexv 3172	. . . 4 |- ( E. g e. X ( g .(+) A ) = B <-> E. h e. X ( h .(+) A ) = B )
8	4, 7	syl6bb 276	. . 3 |- ( ( x = A /\ y = B ) -> ( E. g e. X ( g .(+) x ) = y <-> E. h e. X ( h .(+) A ) = B ) )
9		gaorb.1	. . . 4 |- .~ = { <. x , y >. | ( { x , y } C_ Y /\ E. g e. X ( g .(+) x ) = y ) }
10		vex 3203	. . . . . . 7 |- x e. _V
11		vex 3203	. . . . . . 7 |- y e. _V
12	10, 11	prss 4351	. . . . . 6 |- ( ( x e. Y /\ y e. Y ) <-> { x , y } C_ Y )
13	12	anbi1i 731	. . . . 5 |- ( ( ( x e. Y /\ y e. Y ) /\ E. g e. X ( g .(+) x ) = y ) <-> ( { x , y } C_ Y /\ E. g e. X ( g .(+) x ) = y ) )
14	13	opabbii 4717	. . . 4 |- { <. x , y >. | ( ( x e. Y /\ y e. Y ) /\ E. g e. X ( g .(+) x ) = y ) } = { <. x , y >. | ( { x , y } C_ Y /\ E. g e. X ( g .(+) x ) = y ) }
15	9, 14	eqtr4i 2647	. . 3 |- .~ = { <. x , y >. | ( ( x e. Y /\ y e. Y ) /\ E. g e. X ( g .(+) x ) = y ) }
16	8, 15	brab2a 5194	. 2 |- ( A .~ B <-> ( ( A e. Y /\ B e. Y ) /\ E. h e. X ( h .(+) A ) = B ) )
17		df-3an 1039	. 2 |- ( ( A e. Y /\ B e. Y /\ E. h e. X ( h .(+) A ) = B ) <-> ( ( A e. Y /\ B e. Y ) /\ E. h e. X ( h .(+) A ) = B ) )
18	16, 17	bitr4i 267	1 |- ( A .~ B <-> ( A e. Y /\ B e. Y /\ E. h e. X ( h .(+) A ) = B ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   {cpr 4179   class class class wbr 4653   {copab 4712  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  gaorber  17741  orbsta  17746  sylow2alem1  18032  sylow2alem2  18033  sylow3lem3  18044