Theorem isps 9988

Description: The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation.

Assertion

Ref	Expression
isps	|- (R e. A -> (R e. Poset <-> (Rel R /\ (R o. R) C_ R /\ (R i^i `'R) = ( _I |` U.U.R))))

Proof of Theorem isps

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		id 73	. . . 4 |- (r = R -> r = R)
6	4, 5	sseq12d 2646	. . 3 |- (r = R -> ((r o. r) C_ r <-> (R o. R) C_ R))
7		cnveq 4135	. . . . 5 |- (r = R -> `'r = `'R)
8	5, 7	ineq12d 2797	. . . 4 |- (r = R -> (r i^i `'r) = (R i^i `'R))
9		unieq 3185	. . . . . 6 |- (r = R -> U.r = U.R)
10	9	unieqd 3188	. . . . 5 |- (r = R -> U.U.r = U.U.R)
11		reseq2 4219	. . . . 5 |- (U.U.r = U.U.R -> ( _I |` U.U.r) = ( _I |` U.U.R))
12	10, 11	syl 12	. . . 4 |- (r = R -> ( _I |` U.U.r) = ( _I |` U.U.R))
13	8, 12	eqeq12d 1899	. . 3 |- (r = R -> ((r i^i `'r) = ( _I |` U.U.r) <-> (R i^i `'R) = ( _I |` U.U.R)))
14	1, 6, 13	3anbi123d 1168	. 2 |- (r = R -> ((Rel r /\ (r o. r) C_ r /\ (r i^i `'r) = ( _I |` U.U.r)) <-> (Rel R /\ (R o. R) C_ R /\ (R i^i `'R) = ( _I |` U.U.R))))
15		df-ps 9984	. 2 |- Poset = {r | (Rel r /\ (r o. r) C_ r /\ (r i^i `'r) = ( _I |` U.U.r))}
16	14, 15	elab2g 2406	1 |- (R e. A -> (R e. Poset <-> (Rel R /\ (R o. R) C_ R /\ (R i^i `'R) = ( _I |` U.U.R))))

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858   = wceq 1298   e. wcel 1300   i^i cin 2592   C_ wss 2593  U.cuni 3177   _I cid 3582  `'ccnv 3985   |` cres 3988   o. ccom 3990  Rel wrel 3991  Posetcps 9980

This theorem is referenced by:  psrel 9989  pslem 9990  posanref 10186  postr 10187  empos 14583  dupos1 14586  inposet 14620  dispos 14632  pospos 14635  lteqtpos 15024

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-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-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-res 4006  df-ps 9984

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