Theorem grplactfval 8080

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 3965	. . . . . . 7 |- (g = A -> (gGa) = (AGa))
2	1	eqeq2d 1485	. . . . . 6 |- (g = A -> (b = (gGa) <-> b = (AGa)))
3	2	anbi2d 615	. . . . 5 |- (g = A -> ((a e. X /\ b = (gGa)) <-> (a e. X /\ b = (AGa))))
4	3	opabbidv 2667	. . . 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 3776	. . 3 |- ((A e. X /\ {<.a, b>. | (a e. X /\ b = (AGa))} e. V) -> (F` A) = {<.a, b>. | (a e. X /\ b = (AGa))})
7		rnexg 3356	. . . . 5 |- (G e. Grp -> ran G e. V)
8		grplact.2	. . . . 5 |- X = ran G
9	7, 8	syl5eqel 1551	. . . 4 |- (G e. Grp -> X e. V)
10		opabex2g 3608	. . . 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 451	. 2 |- ((A e. X /\ G e. Grp) -> (F` A) = {<.a, b>. | (a e. X /\ b = (AGa))})
13	12	ancoms 436	1 |- ((G e. Grp /\ A e. X) -> (F` A) = {<.a, b>. | (a e. X /\ b = (AGa))})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  Vcvv 1809  {copab 2663  ran crn 3168  ` cfv 3179  (class class class)co 3960  Grpcgr 8016

This theorem is referenced by:  grplactval 8081  grplactf1o 8082  cayleylem2 10401

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-fv 3195  df-opr 3962

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