Theorem isps 17202

Description: The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation. (Contributed by NM, 11-May-2008.)

Assertion

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

Proof of Theorem isps

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		releq 5201	. . 3 |- ( r = R -> ( Rel r <-> Rel R ) )
2		coeq1 5279	. . . . 5 |- ( r = R -> ( r o. r ) = ( R o. r ) )
3		coeq2 5280	. . . . 5 |- ( r = R -> ( R o. r ) = ( R o. R ) )
4	2, 3	eqtrd 2656	. . . 4 |- ( r = R -> ( r o. r ) = ( R o. R ) )
5		id 22	. . . 4 |- ( r = R -> r = R )
6	4, 5	sseq12d 3634	. . 3 |- ( r = R -> ( ( r o. r ) C_ r <-> ( R o. R ) C_ R ) )
7		cnveq 5296	. . . . 5 |- ( r = R -> `' r = `' R )
8	5, 7	ineq12d 3815	. . . 4 |- ( r = R -> ( r i^i `' r ) = ( R i^i `' R ) )
9		unieq 4444	. . . . . 6 |- ( r = R -> U. r = U. R )
10	9	unieqd 4446	. . . . 5 |- ( r = R -> U. U. r = U. U. R )
11	10	reseq2d 5396	. . . 4 |- ( r = R -> ( _I |` U. U. r ) = ( _I |` U. U. R ) )
12	8, 11	eqeq12d 2637	. . 3 |- ( r = R -> ( ( r i^i `' r ) = ( _I |` U. U. r ) <-> ( R i^i `' R ) = ( _I |` U. U. R ) ) )
13	1, 6, 12	3anbi123d 1399	. 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 ) ) ) )
14		df-ps 17200	. 2 |- PosetRel = { r | ( Rel r /\ ( r o. r ) C_ r /\ ( r i^i `' r ) = ( _I |` U. U. r ) ) }
15	13, 14	elab2g 3353	1 |- ( R e. A -> ( R e. PosetRel <-> ( Rel R /\ ( R o. R ) C_ R /\ ( R i^i `' R ) = ( _I |` U. U. R ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   U.cuni 4436   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   Rel wrel 5119   PosetRelcps 17198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-res 5126  df-ps 17200

This theorem is referenced by:  psrel  17203  psref2  17204  pstr2  17205  cnvps  17212  psss  17214  letsr  17227