Theorem cnvps 15716

Description: The converse of a poset is a poset. In the general case ( `' R e. PosetRel -> R e. PosetRel ) is not true. See cnvpsb 15717 for a special case where the property holds. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
cnvps	|- ( R e. PosetRel -> `' R e. PosetRel )

Proof of Theorem cnvps

Step	Hyp	Ref	Expression
1		relcnv 5380	. . 3 |- Rel `' R
2	1	a1i 11	. 2 |- ( R e. PosetRel -> Rel `' R )
3		cnvco 5194	. . 3 |- `' ( R o. R ) = ( `' R o. `' R )
4		pstr2 15709	. . . 4 |- ( R e. PosetRel -> ( R o. R ) C_ R )
5		cnvss 5181	. . . 4 |- ( ( R o. R ) C_ R -> `' ( R o. R ) C_ `' R )
6	4, 5	syl 16	. . 3 |- ( R e. PosetRel -> `' ( R o. R ) C_ `' R )
7	3, 6	syl5eqssr 3554	. 2 |- ( R e. PosetRel -> ( `' R o. `' R ) C_ `' R )
8		psrel 15707	. . . . . 6 |- ( R e. PosetRel -> Rel R )
9		dfrel2 5463	. . . . . 6 |- ( Rel R <-> `' `' R = R )
10	8, 9	sylib 196	. . . . 5 |- ( R e. PosetRel -> `' `' R = R )
11	10	ineq2d 3705	. . . 4 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( `' R i^i R ) )
12		incom 3696	. . . 4 |- ( `' R i^i R ) = ( R i^i `' R )
13	11, 12	syl6eq 2524	. . 3 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( R i^i `' R ) )
14		psref2 15708	. . 3 |- ( R e. PosetRel -> ( R i^i `' R ) = ( _I |` U. U. R ) )
15		relcnvfld 5544	. . . . 5 |- ( Rel R -> U. U. R = U. U. `' R )
16	8, 15	syl 16	. . . 4 |- ( R e. PosetRel -> U. U. R = U. U. `' R )
17	16	reseq2d 5279	. . 3 |- ( R e. PosetRel -> ( _I |` U. U. R ) = ( _I |` U. U. `' R ) )
18	13, 14, 17	3eqtrd 2512	. 2 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( _I |` U. U. `' R ) )
19		cnvexg 6741	. . 3 |- ( R e. PosetRel -> `' R e. _V )
20		isps 15706	. . 3 |- ( `' R e. _V -> ( `' R e. PosetRel <-> ( Rel `' R /\ ( `' R o. `' R ) C_ `' R /\ ( `' R i^i `' `' R ) = ( _I |` U. U. `' R ) ) ) )
21	19, 20	syl 16	. 2 |- ( R e. PosetRel -> ( `' R e. PosetRel <-> ( Rel `' R /\ ( `' R o. `' R ) C_ `' R /\ ( `' R i^i `' `' R ) = ( _I |` U. U. `' R ) ) ) )
22	2, 7, 18, 21	mpbir3and 1179	1 |- ( R e. PosetRel -> `' R e. PosetRel )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 973   = wceq 1379   e. wcel 1767   _Vcvv 3118   i^i cin 3480   C_ wss 3481   U.cuni 4251   _I cid 4796   `'ccnv 5004   |` cres 5007   o. ccom 5009   Rel wrel 5010   PosetRelcps 15702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ps 15704

This theorem is referenced by:  cnvpsb  15717  cnvtsr  15726  ordtcnv  19570  xrge0iifhmeo  27734