Theorem istoset 10209

Description: The predicate is a toset. (Contributed by FL, 3-Nov-2009.)

Hypothesis

Ref	Expression
istoset.1	|- X = U.U.R

Assertion

Ref	Expression
istoset	|- (R e. A -> (R e. Toset <-> (R e. Poset /\ A.x e. X A.y e. X (xRy \/ yRx))))

Distinct variable groups:   x,R,y   x,X,y

Proof of Theorem istoset

Step	Hyp	Ref	Expression
1		unieq 3185	. . . . . . . 8 |- (r = R -> U.r = U.R)
2	1	unieqd 3188	. . . . . . 7 |- (r = R -> U.U.r = U.U.R)
3		breq 3340	. . . . . . . . 9 |- (r = R -> (xry <-> xRy))
4		breq 3340	. . . . . . . . 9 |- (r = R -> (yrx <-> yRx))
5	3, 4	orbi12d 689	. . . . . . . 8 |- (r = R -> ((xry \/ yrx) <-> (xRy \/ yRx)))
6	2, 5	raleqbidv 2274	. . . . . . 7 |- (r = R -> (A.y e. U.U.r(xry \/ yrx) <-> A.y e. U.U.R(xRy \/ yRx)))
7	2, 6	raleqbidv 2274	. . . . . 6 |- (r = R -> (A.x e. U.U.rA.y e. U.U.r(xry \/ yrx) <-> A.x e. U.U.RA.y e. U.U.R(xRy \/ yRx)))
8		istoset.1	. . . . . . . . 9 |- X = U.U.R
9	8	eqcomi 1888	. . . . . . . 8 |- U.U.R = X
10	9	a1i 8	. . . . . . 7 |- (r = R -> U.U.R = X)
11	10	raleqdv 2269	. . . . . . 7 |- (r = R -> (A.y e. U.U.R(xRy \/ yRx) <-> A.y e. X (xRy \/ yRx)))
12	10, 11	raleqbidv 2274	. . . . . 6 |- (r = R -> (A.x e. U.U.RA.y e. U.U.R(xRy \/ yRx) <-> A.x e. X A.y e. X (xRy \/ yRx)))
13	7, 12	bitrd 587	. . . . 5 |- (r = R -> (A.x e. U.U.rA.y e. U.U.r(xry \/ yrx) <-> A.x e. X A.y e. X (xRy \/ yRx)))
14	13	elabg 2405	. . . 4 |- (R e. A -> (R e. {r | A.x e. U.U.rA.y e. U.U.r(xry \/ yrx)} <-> A.x e. X A.y e. X (xRy \/ yRx)))
15	14	anbi2d 678	. . 3 |- (R e. A -> ((R e. Poset /\ R e. {r | A.x e. U.U.rA.y e. U.U.r(xry \/ yrx)}) <-> (R e. Poset /\ A.x e. X A.y e. X (xRy \/ yRx))))
16		elin 2786	. . 3 |- (R e. (Poset i^i {r | A.x e. U.U.rA.y e. U.U.r(xry \/ yrx)}) <-> (R e. Poset /\ R e. {r | A.x e. U.U.rA.y e. U.U.r(xry \/ yrx)}))
17	15, 16	syl5bb 591	. 2 |- (R e. A -> (R e. (Poset i^i {r | A.x e. U.U.rA.y e. U.U.r(xry \/ yrx)}) <-> (R e. Poset /\ A.x e. X A.y e. X (xRy \/ yRx))))
18		df-toset 10208	. . 3 |- Toset = (Poset i^i {r | A.x e. U.U.rA.y e. U.U.r(xry \/ yrx)})
19	18	eleq2i 1961	. 2 |- (R e. Toset <-> R e. (Poset i^i {r | A.x e. U.U.rA.y e. U.U.r(xry \/ yrx)}))
20	17, 19	syl5bb 591	1 |- (R e. A -> (R e. Toset <-> (R e. Poset /\ A.x e. X A.y e. X (xRy \/ yRx))))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105   i^i cin 2592  U.cuni 3177   class class class wbr 3338  Posetcps 9980   Toset ccha 10207

This theorem is referenced by:  tosdir 10358  dutos1 14626  istoset2 14628  tostos 14637

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-in 2603  df-uni 3178  df-br 3339  df-toset 10208

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