Theorem grprlidrid 9337

Description: In a group a left and right identity element is a left identity element. (Contributed by FL, 5-Feb-2010.)

Hypothesis

Ref	Expression
grprlidrid.1	|- X = ran G

Assertion

Ref	Expression
grprlidrid	|- (G e. Grp -> U.{u e. X | A.x e. X ((uGx) = x /\ (xGu) = x)} = U.{u e. X | A.x e. X (uGx) = x})

Distinct variable groups:   u,G,x   u,X,x

Proof of Theorem grprlidrid

Step	Hyp	Ref	Expression
1		simpl 346	. . . . . . 7 |- (((uGx) = x /\ (xGu) = x) -> (uGx) = x)
2	1	a1i 8	. . . . . 6 |- ((G e. Grp /\ x e. X) -> (((uGx) = x /\ (xGu) = x) -> (uGx) = x))
3	2	ralimdvaa 2171	. . . . 5 |- (G e. Grp -> (A.x e. X ((uGx) = x /\ (xGu) = x) -> A.x e. X (uGx) = x))
4	3	a1d 15	. . . 4 |- (G e. Grp -> (u e. X -> (A.x e. X ((uGx) = x /\ (xGu) = x) -> A.x e. X (uGx) = x)))
5	4	r19.21aiv 2175	. . 3 |- (G e. Grp -> A.u e. X (A.x e. X ((uGx) = x /\ (xGu) = x) -> A.x e. X (uGx) = x))
6		ssid 2634	. . 3 |- X C_ X
7	5, 6	jctil 316	. 2 |- (G e. Grp -> (X C_ X /\ A.u e. X (A.x e. X ((uGx) = x /\ (xGu) = x) -> A.x e. X (uGx) = x)))
8		grprlidrid.1	. . . 4 |- X = ran G
9	8	grpidinv 9332	. . 3 |- (G e. Grp -> E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)))
10		simpl 346	. . . . 5 |- ((((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> ((uGx) = x /\ (xGu) = x))
11	10	ralimi 2168	. . . 4 |- (A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> A.x e. X ((uGx) = x /\ (xGu) = x))
12	11	reximi 2198	. . 3 |- (E.u e. X A.x e. X (((uGx) = x /\ (xGu) = x) /\ E.y e. X ((yGx) = u /\ (xGy) = u)) -> E.u e. X A.x e. X ((uGx) = x /\ (xGu) = x))
13	9, 12	syl 12	. 2 |- (G e. Grp -> E.u e. X A.x e. X ((uGx) = x /\ (xGu) = x))
14	8	grpideu 9333	. 2 |- (G e. Grp -> E!u e. X A.x e. X (uGx) = x)
15		reuuniss2 3817	. 2 |- (((X C_ X /\ A.u e. X (A.x e. X ((uGx) = x /\ (xGu) = x) -> A.x e. X (uGx) = x)) /\ (E.u e. X A.x e. X ((uGx) = x /\ (xGu) = x) /\ E!u e. X A.x e. X (uGx) = x)) -> U.{u e. X | A.x e. X ((uGx) = x /\ (xGu) = x)} = U.{u e. X | A.x e. X (uGx) = x})
16	7, 13, 14, 15	syl12anc 1098	1 |- (G e. Grp -> U.{u e. X | A.x e. X ((uGx) = x /\ (xGu) = x)} = U.{u e. X | A.x e. X (uGx) = x})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107  {crab 2108   C_ wss 2593  U.cuni 3177  ran crn 3987  (class class class)co 4884  Grpcgr 9311

This theorem is referenced by:  grpidvallem 9341  grpidval 9342

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-grp 9316

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