Theorem isps 8641

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) (_ R /\ (R i^i `'R) = (I |` U.U.R))))

Proof of Theorem isps

Step	Hyp	Ref	Expression
1		releq 3249	. . 3 |- (r = R -> (Rel r <-> Rel R))
2		coeq1 3287	. . . . 5 |- (r = R -> (r o. r) = (R o. r))
3		coeq2 3288	. . . . 5 |- (r = R -> (R o. r) = (R o. R))
4	2, 3	eqtrd 1510	. . . 4 |- (r = R -> (r o. r) = (R o. R))
5		id 59	. . . 4 |- (r = R -> r = R)
6	4, 5	sseq12d 2093	. . 3 |- (r = R -> ((r o. r) (_ r <-> (R o. R) (_ R))
7		cnveq 3298	. . . . 5 |- (r = R -> `'r = `'R)
8	5, 7	ineq12d 2221	. . . 4 |- (r = R -> (r i^i `'r) = (R i^i `'R))
9		unieq 2514	. . . . . 6 |- (r = R -> U.r = U.R)
10	9	unieqd 2516	. . . . 5 |- (r = R -> U.U.r = U.U.R)
11		reseq2 3375	. . . . 5 |- (U.U.r = U.U.R -> (I |` U.U.r) = (I |` U.U.R))
12	10, 11	syl 10	. . . 4 |- (r = R -> (I |` U.U.r) = (I |` U.U.R))
13	8, 12	eqeq12d 1492	. . 3 |- (r = R -> ((r i^i `'r) = (I |` U.U.r) <-> (R i^i `'R) = (I |` U.U.R)))
14	1, 6, 13	3anbi123d 895	. 2 |- (r = R -> ((Rel r /\ (r o. r) (_ r /\ (r i^i `'r) = (I |` U.U.r)) <-> (Rel R /\ (R o. R) (_ R /\ (R i^i `'R) = (I |` U.U.R))))
15		df-ps 8635	. 2 |- Poset = {r | (Rel r /\ (r o. r) (_ r /\ (r i^i `'r) = (I |` U.U.r))}
16	14, 15	elab2g 1903	1 |- (R e. A -> (R e. Poset <-> (Rel R /\ (R o. R) (_ R /\ (R i^i `'R) = (I |` U.U.R))))

Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 777   = wceq 958   e. wcel 960   i^i cin 2049   (_ wss 2050  U.cuni 2507  Icid 2837  `'ccnv 3175   |` cres 3178   o. ccom 3180  Rel wrel 3181  Posetcps 8629

This theorem is referenced by:  psrel 8642  pslem 8643  inposet 10477

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-res 3196  df-ps 8635

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