Theorem grplactfval 8092

Description: The left group action of element A of group G. (Contributed by Paul Chapman, 18-Mar-2008.)

Hypotheses

Ref	Expression
grplact.1	|- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}
grplact.2	|- X = ran G

Assertion

Ref	Expression
grplactfval	|- ((G e. Grp /\ A e. X) -> (F` A) = {<.a, b>. | (a e. X /\ b = (AGa))})

Distinct variable groups:   A,a,b,g,h   G,a,b,g,h   X,a,b,g,h

Proof of Theorem grplactfval

Step	Hyp	Ref	Expression
1		opreq1 3974	. . . . . . 7 |- (g = A -> (gGa) = (AGa))
2	1	eqeq2d 1489	. . . . . 6 |- (g = A -> (b = (gGa) <-> b = (AGa)))
3	2	anbi2d 618	. . . . 5 |- (g = A -> ((a e. X /\ b = (gGa)) <-> (a e. X /\ b = (AGa))))
4	3	opabbidv 2675	. . . 4 |- (g = A -> {<.a, b>. | (a e. X /\ b = (gGa))} = {<.a, b>. | (a e. X /\ b = (AGa))})
5		grplact.1	. . . 4 |- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}
6	4, 5	fvopab4g 3785	. . 3 |- ((A e. X /\ {<.a, b>. | (a e. X /\ b = (AGa))} e. V) -> (F` A) = {<.a, b>. | (a e. X /\ b = (AGa))})
7		rnexg 3365	. . . . 5 |- (G e. Grp -> ran G e. V)
8		grplact.2	. . . . 5 |- X = ran G
9	7, 8	syl5eqel 1555	. . . 4 |- (G e. Grp -> X e. V)
10		opabex2g 3617	. . . 4 |- (X e. V -> {<.a, b>. | (a e. X /\ b = (AGa))} e. V)
11	9, 10	syl 10	. . 3 |- (G e. Grp -> {<.a, b>. | (a e. X /\ b = (AGa))} e. V)
12	6, 11	sylan2 453	. 2 |- ((A e. X /\ G e. Grp) -> (F` A) = {<.a, b>. | (a e. X /\ b = (AGa))})
13	12	ancoms 438	1 |- ((G e. Grp /\ A e. X) -> (F` A) = {<.a, b>. | (a e. X /\ b = (AGa))})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  {copab 2671  ran crn 3177  ` cfv 3188  (class class class)co 3969  Grpcgr 8030

This theorem is referenced by:  grplactval 8093  grplactf1o 8094  cayleylem2 10405

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-opr 3971

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