Theorem isexid 10364

Description: The predicate G has a left and right identity element. (Contributed by FL, 2-Nov-2009.)

Hypothesis

Ref	Expression
isexid.1	|- X = dom dom G

Assertion

Ref	Expression
isexid	|- (G e. A -> (G e. ExId <-> E.x e. X A.y e. X ((yGx) = y /\ (xGy) = y)))

Distinct variable group:   x,G,y

Proof of Theorem isexid

Step	Hyp	Ref	Expression
1		dmeq 4157	. . . . 5 |- (g = G -> dom g = dom G)
2	1	dmeqd 4159	. . . 4 |- (g = G -> dom dom g = dom dom G)
3		opreq 4888	. . . . . . 7 |- (g = G -> (ygx) = (yGx))
4	3	eqeq1d 1892	. . . . . 6 |- (g = G -> ((ygx) = y <-> (yGx) = y))
5		opreq 4888	. . . . . . 7 |- (g = G -> (xgy) = (xGy))
6	5	eqeq1d 1892	. . . . . 6 |- (g = G -> ((xgy) = y <-> (xGy) = y))
7	4, 6	anbi12d 690	. . . . 5 |- (g = G -> (((ygx) = y /\ (xgy) = y) <-> ((yGx) = y /\ (xGy) = y)))
8	2, 7	raleqbidv 2274	. . . 4 |- (g = G -> (A.y e. dom dom g((ygx) = y /\ (xgy) = y) <-> A.y e. dom dom G((yGx) = y /\ (xGy) = y)))
9	2, 8	rexeqbidv 2275	. . 3 |- (g = G -> (E.x e. dom dom gA.y e. dom dom g((ygx) = y /\ (xgy) = y) <-> E.x e. dom dom GA.y e. dom dom G((yGx) = y /\ (xGy) = y)))
10		isexid.1	. . . . . . 7 |- X = dom dom G
11	10	eqcomi 1888	. . . . . 6 |- dom dom G = X
12	11	eleq2i 1961	. . . . 5 |- (x e. dom dom G <-> x e. X)
13	11	eleq2i 1961	. . . . . . 7 |- (y e. dom dom G <-> y e. X)
14	13	imbi1i 203	. . . . . 6 |- ((y e. dom dom G -> ((yGx) = y /\ (xGy) = y)) <-> (y e. X -> ((yGx) = y /\ (xGy) = y)))
15	14	ralbii2 2131	. . . . 5 |- (A.y e. dom dom G((yGx) = y /\ (xGy) = y) <-> A.y e. X ((yGx) = y /\ (xGy) = y))
16	12, 15	anbi12i 540	. . . 4 |- ((x e. dom dom G /\ A.y e. dom dom G((yGx) = y /\ (xGy) = y)) <-> (x e. X /\ A.y e. X ((yGx) = y /\ (xGy) = y)))
17	16	rexbii2 2132	. . 3 |- (E.x e. dom dom GA.y e. dom dom G((yGx) = y /\ (xGy) = y) <-> E.x e. X A.y e. X ((yGx) = y /\ (xGy) = y))
18	9, 17	syl6bb 595	. 2 |- (g = G -> (E.x e. dom dom gA.y e. dom dom g((ygx) = y /\ (xgy) = y) <-> E.x e. X A.y e. X ((yGx) = y /\ (xGy) = y)))
19		df-exid 10362	. 2 |- ExId = {g | E.x e. dom dom gA.y e. dom dom g((ygx) = y /\ (xgy) = y)}
20	18, 19	elab2g 2406	1 |- (G e. A -> (G e. ExId <-> E.x e. X A.y e. X ((yGx) = y /\ (xGy) = y)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  dom cdm 3986  (class class class)co 4884   ExId cexid 10361

This theorem is referenced by:  opidon 10369  isexid2 10372  ismnd 10390  symgfo 14730  exidres 16031

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-v 2294  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-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-exid 10362

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