Theorem isps 8645

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 3243	. . 3 |- (r = R -> (Rel r <-> Rel R))
2		coeq1 3281	. . . . 5 |- (r = R -> (r o. r) = (R o. r))
3		coeq2 3282	. . . . 5 |- (r = R -> (R o. r) = (R o. R))
4	2, 3	eqtrd 1507	. . . 4 |- (r = R -> (r o. r) = (R o. R))
5		id 59	. . . 4 |- (r = R -> r = R)
6	4, 5	sseq12d 2090	. . 3 |- (r = R -> ((r o. r) (_ r <-> (R o. R) (_ R))
7		cnveq 3292	. . . . 5 |- (r = R -> `'r = `'R)
8	5, 7	ineq12d 2218	. . . 4 |- (r = R -> (r i^i `'r) = (R i^i `'R))
9		unieq 2510	. . . . . 6 |- (r = R -> U.r = U.R)
10	9	unieqd 2512	. . . . 5 |- (r = R -> U.U.r = U.U.R)
11		reseq2 3369	. . . . 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 1489	. . 3 |- (r = R -> ((r i^i `'r) = (I |` U.U.r) <-> (R i^i `'R) = (I |` U.U.R)))
14	1, 6, 13	3anbi123d 893	. 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 8639	. 2 |- Poset = {r | (Rel r /\ (r o. r) (_ r /\ (r i^i `'r) = (I |` U.U.r))}
16	14, 15	elab2g 1900	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 775   = wceq 956   e. wcel 958   i^i cin 2046   (_ wss 2047  U.cuni 2503  Icid 2831  `'ccnv 3169   |` cres 3172   o. ccom 3174  Rel wrel 3175  Posetcps 8633

This theorem is referenced by:  psrel 8646  pslem 8647

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-res 3190  df-ps 8639

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