Theorem psref2 17204

Description: A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)

Assertion

Ref	Expression
psref2	|- ( R e. PosetRel -> ( R i^i `' R ) = ( _I |` U. U. R ) )

Proof of Theorem psref2

Step	Hyp	Ref	Expression
1		isps 17202	. . 3 |- ( R e. PosetRel -> ( R e. PosetRel <-> ( Rel R /\ ( R o. R ) C_ R /\ ( R i^i `' R ) = ( _I |` U. U. R ) ) ) )
2	1	ibi 256	. 2 |- ( R e. PosetRel -> ( Rel R /\ ( R o. R ) C_ R /\ ( R i^i `' R ) = ( _I |` U. U. R ) ) )
3	2	simp3d 1075	1 |- ( R e. PosetRel -> ( R i^i `' R ) = ( _I |` U. U. R ) )

Syntax hints:   -> wi 4   /\ 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:  pslem  17206  cnvps  17212  tsrdir  17238