Theorem isprs 14565

Description: The predicate "is a preset".

Assertion

Ref	Expression
isprs	|- (R e. A -> (R e. Preset <-> (Rel R /\ (R o. R) C_ R /\ ( _I |` U.U.R) C_ R)))

Proof of Theorem isprs

Step	Hyp	Ref	Expression
1		releq 4071	. . 3 |- (r = R -> (Rel r <-> Rel R))
2		coeq1 4123	. . . . 5 |- (r = R -> (r o. r) = (R o. r))
3		coeq2 4124	. . . . 5 |- (r = R -> (R o. r) = (R o. R))
4	2, 3	eqtrd 1925	. . . 4 |- (r = R -> (r o. r) = (R o. R))
5		sseq12 2640	. . . 4 |- (((r o. r) = (R o. R) /\ r = R) -> ((r o. r) C_ r <-> (R o. R) C_ R))
6	4, 5	mpancom 769	. . 3 |- (r = R -> ((r o. r) C_ r <-> (R o. R) C_ R))
7		unieq 3185	. . . . . 6 |- (r = R -> U.r = U.R)
8	7	unieqd 3188	. . . . 5 |- (r = R -> U.U.r = U.U.R)
9		reseq2 4219	. . . . 5 |- (U.U.r = U.U.R -> ( _I |` U.U.r) = ( _I |` U.U.R))
10	8, 9	syl 12	. . . 4 |- (r = R -> ( _I |` U.U.r) = ( _I |` U.U.R))
11		sseq12 2640	. . . 4 |- ((( _I |` U.U.r) = ( _I |` U.U.R) /\ r = R) -> (( _I |` U.U.r) C_ r <-> ( _I |` U.U.R) C_ R))
12	10, 11	mpancom 769	. . 3 |- (r = R -> (( _I |` U.U.r) C_ r <-> ( _I |` U.U.R) C_ R))
13	1, 6, 12	3anbi123d 1168	. 2 |- (r = R -> ((Rel r /\ (r o. r) C_ r /\ ( _I |` U.U.r) C_ r) <-> (Rel R /\ (R o. R) C_ R /\ ( _I |` U.U.R) C_ R)))
14		df-prs 14563	. 2 |- Preset = {r | (Rel r /\ (r o. r) C_ r /\ ( _I |` U.U.r) C_ r)}
15	13, 14	elab2g 2406	1 |- (R e. A -> (R e. Preset <-> (Rel R /\ (R o. R) C_ R /\ ( _I |` U.U.R) C_ R)))

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858   = wceq 1298   e. wcel 1300   C_ wss 2593  U.cuni 3177   _I cid 3582   |` cres 3988   o. ccom 3990  Rel wrel 3991   Preset cpreset 14555

This theorem is referenced by:  preorel 14566  preodom2 14567  preoref12 14569  preoran2 14571  preotr1 14575  altprs2 14577  int2pre 14578  sqpre 14579  dupre1 14584

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-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-v 2294  df-in 2603  df-ss 2605  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-co 4003  df-res 4006  df-prs 14563

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