Theorem istoset2 14628

Description: The predicate is a toset. dom R is preferred to U.U.R to express the underling set.

Hypothesis

Ref	Expression
istoset2.1	|- X = dom R

Assertion

Ref	Expression
istoset2	|- (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 istoset2

Step	Hyp	Ref	Expression
1		tosdir 10358	. . . . 5 |- Toset C_ Dir
2	1	sseli 2617	. . . 4 |- (R e. Toset -> R e. Dir)
3		dirdm 10354	. . . 4 |- (R e. Dir -> dom R = U.U.R)
4	2, 3	syl 12	. . 3 |- (R e. Toset -> dom R = U.U.R)
5		istoset2.1	. . . 4 |- X = dom R
6		eqtr 1904	. . . . 5 |- ((X = dom R /\ dom R = U.U.R) -> X = U.U.R)
7		id 73	. . . . . . . 8 |- (X = U.U.R -> X = U.U.R)
8		raleq 2266	. . . . . . . 8 |- (X = U.U.R -> (A.y e. X (xRy \/ yRx) <-> A.y e. U.U.R(xRy \/ yRx)))
9	7, 8	raleqbidv 2274	. . . . . . 7 |- (X = U.U.R -> (A.x e. X A.y e. X (xRy \/ yRx) <-> A.x e. U.U.RA.y e. U.U.R(xRy \/ yRx)))
10	9	anbi2d 678	. . . . . 6 |- (X = U.U.R -> ((R e. Poset /\ A.x e. X A.y e. X (xRy \/ yRx)) <-> (R e. Poset /\ A.x e. U.U.RA.y e. U.U.R(xRy \/ yRx))))
11		eqid 1884	. . . . . . . 8 |- U.U.R = U.U.R
12	11	istoset 10209	. . . . . . 7 |- (R e. Toset -> (R e. Toset <-> (R e. Poset /\ A.x e. U.U.RA.y e. U.U.R(xRy \/ yRx))))
13	12	ibi 652	. . . . . 6 |- (R e. Toset -> (R e. Poset /\ A.x e. U.U.RA.y e. U.U.R(xRy \/ yRx)))
14	10, 13	syl5bir 227	. . . . 5 |- (X = U.U.R -> (R e. Toset -> (R e. Poset /\ A.x e. X A.y e. X (xRy \/ yRx))))
15	6, 14	syl 12	. . . 4 |- ((X = dom R /\ dom R = U.U.R) -> (R e. Toset -> (R e. Poset /\ A.x e. X A.y e. X (xRy \/ yRx))))
16	5, 15	mpan 759	. . 3 |- (dom R = U.U.R -> (R e. Toset -> (R e. Poset /\ A.x e. X A.y e. X (xRy \/ yRx))))
17	4, 16	mpcom 60	. 2 |- (R e. Toset -> (R e. Poset /\ A.x e. X A.y e. X (xRy \/ yRx)))
18		posispre 14582	. . . . 5 |- (R e. Poset -> R e. Preset )
19		preodom2 14567	. . . . 5 |- (R e. Preset -> dom R = U.U.R)
20		biidd 188	. . . . . . . . . . 11 |- (X = U.U.R -> ((xRy \/ yRx) <-> (xRy \/ yRx)))
21	7, 20	raleqbidv 2274	. . . . . . . . . 10 |- (X = U.U.R -> (A.y e. X (xRy \/ yRx) <-> A.y e. U.U.R(xRy \/ yRx)))
22	7, 21	raleqbidv 2274	. . . . . . . . 9 |- (X = U.U.R -> (A.x e. X A.y e. X (xRy \/ yRx) <-> A.x e. U.U.RA.y e. U.U.R(xRy \/ yRx)))
23	22	imbi1d 675	. . . . . . . 8 |- (X = U.U.R -> ((A.x e. X A.y e. X (xRy \/ yRx) -> R e. Toset ) <-> (A.x e. U.U.RA.y e. U.U.R(xRy \/ yRx) -> R e. Toset )))
24	11	istoset 10209	. . . . . . . . . . 11 |- (R e. Poset -> (R e. Toset <-> (R e. Poset /\ A.x e. U.U.RA.y e. U.U.R(xRy \/ yRx))))
25	24	biimprd 171	. . . . . . . . . 10 |- (R e. Poset -> ((R e. Poset /\ A.x e. U.U.RA.y e. U.U.R(xRy \/ yRx)) -> R e. Toset ))
26	25	exp3a 405	. . . . . . . . 9 |- (R e. Poset -> (R e. Poset -> (A.x e. U.U.RA.y e. U.U.R(xRy \/ yRx) -> R e. Toset )))
27	26	pm2.43i 78	. . . . . . . 8 |- (R e. Poset -> (A.x e. U.U.RA.y e. U.U.R(xRy \/ yRx) -> R e. Toset ))
28	23, 27	syl5bir 227	. . . . . . 7 |- (X = U.U.R -> (R e. Poset -> (A.x e. X A.y e. X (xRy \/ yRx) -> R e. Toset )))
29	6, 28	syl 12	. . . . . 6 |- ((X = dom R /\ dom R = U.U.R) -> (R e. Poset -> (A.x e. X A.y e. X (xRy \/ yRx) -> R e. Toset )))
30	5, 29	mpan 759	. . . . 5 |- (dom R = U.U.R -> (R e. Poset -> (A.x e. X A.y e. X (xRy \/ yRx) -> R e. Toset )))
31	18, 19, 30	3syl 24	. . . 4 |- (R e. Poset -> (R e. Poset -> (A.x e. X A.y e. X (xRy \/ yRx) -> R e. Toset )))
32	31	pm2.43i 78	. . 3 |- (R e. Poset -> (A.x e. X A.y e. X (xRy \/ yRx) -> R e. Toset ))
33	32	imp 377	. 2 |- ((R e. Poset /\ A.x e. X A.y e. X (xRy \/ yRx)) -> R e. Toset )
34	17, 33	impbii 174	1 |- (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  A.wral 2105  U.cuni 3177   class class class wbr 3338  dom cdm 3986  Posetcps 9980   Toset ccha 10207  Dircdir 10348   Preset cpreset 14555

This theorem is referenced by:  tolat 14631

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-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-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-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ps 9984  df-toset 10208  df-dir 10350  df-prs 14563

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