Theorem cnvps 15374

Description: The converse of a poset is a poset. In the general case ( `' R e. PosetRel -> R e. PosetRel ) is not true. See cnvpsb 15375 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 5201	. . 3 |- Rel `' R
2	1	a1i 11	. 2 |- ( R e. PosetRel -> Rel `' R )
3		cnvco 5020	. . 3 |- `' ( R o. R ) = ( `' R o. `' R )
4		pstr2 15367	. . . 4 |- ( R e. PosetRel -> ( R o. R ) C_ R )
5		cnvss 5007	. . . 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 3396	. 2 |- ( R e. PosetRel -> ( `' R o. `' R ) C_ `' R )
8		psrel 15365	. . . . . 6 |- ( R e. PosetRel -> Rel R )
9		dfrel2 5283	. . . . . 6 |- ( Rel R <-> `' `' R = R )
10	8, 9	sylib 196	. . . . 5 |- ( R e. PosetRel -> `' `' R = R )
11	10	ineq2d 3547	. . . 4 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( `' R i^i R ) )
12		incom 3538	. . . 4 |- ( `' R i^i R ) = ( R i^i `' R )
13	11, 12	syl6eq 2486	. . 3 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( R i^i `' R ) )
14		psref2 15366	. . 3 |- ( R e. PosetRel -> ( R i^i `' R ) = ( _I |` U. U. R ) )
15		relcnvfld 5363	. . . . 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 5105	. . 3 |- ( R e. PosetRel -> ( _I |` U. U. R ) = ( _I |` U. U. `' R ) )
18	13, 14, 17	3eqtrd 2474	. 2 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( _I |` U. U. `' R ) )
19		cnvexg 6519	. . 3 |- ( R e. PosetRel -> `' R e. _V )
20		isps 15364	. . 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 1171	1 |- ( R e. PosetRel -> `' R e. PosetRel )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 965   = wceq 1369   e. wcel 1756   _Vcvv 2967   i^i cin 3322   C_ wss 3323   U.cuni 4086   _I cid 4626   `'ccnv 4834   |` cres 4837   o. ccom 4839   Rel wrel 4840   PosetRelcps 15360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ps 15362

This theorem is referenced by:  cnvpsb  15375  cnvtsr  15384  ordtcnv  18780  xrge0iifhmeo  26318