Theorem cnvps 17212

Description: The converse of a poset is a poset. In the general case ( `' R e. PosetRel -> R e. PosetRel ) is not true. See cnvpsb 17213 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 5503	. . 3 |- Rel `' R
2	1	a1i 11	. 2 |- ( R e. PosetRel -> Rel `' R )
3		cnvco 5308	. . 3 |- `' ( R o. R ) = ( `' R o. `' R )
4		pstr2 17205	. . . 4 |- ( R e. PosetRel -> ( R o. R ) C_ R )
5		cnvss 5294	. . . 4 |- ( ( R o. R ) C_ R -> `' ( R o. R ) C_ `' R )
6	4, 5	syl 17	. . 3 |- ( R e. PosetRel -> `' ( R o. R ) C_ `' R )
7	3, 6	syl5eqssr 3650	. 2 |- ( R e. PosetRel -> ( `' R o. `' R ) C_ `' R )
8		psrel 17203	. . . . . 6 |- ( R e. PosetRel -> Rel R )
9		dfrel2 5583	. . . . . 6 |- ( Rel R <-> `' `' R = R )
10	8, 9	sylib 208	. . . . 5 |- ( R e. PosetRel -> `' `' R = R )
11	10	ineq2d 3814	. . . 4 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( `' R i^i R ) )
12		incom 3805	. . . 4 |- ( `' R i^i R ) = ( R i^i `' R )
13	11, 12	syl6eq 2672	. . 3 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( R i^i `' R ) )
14		psref2 17204	. . 3 |- ( R e. PosetRel -> ( R i^i `' R ) = ( _I |` U. U. R ) )
15		relcnvfld 5666	. . . . 5 |- ( Rel R -> U. U. R = U. U. `' R )
16	8, 15	syl 17	. . . 4 |- ( R e. PosetRel -> U. U. R = U. U. `' R )
17	16	reseq2d 5396	. . 3 |- ( R e. PosetRel -> ( _I |` U. U. R ) = ( _I |` U. U. `' R ) )
18	13, 14, 17	3eqtrd 2660	. 2 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( _I |` U. U. `' R ) )
19		cnvexg 7112	. . 3 |- ( R e. PosetRel -> `' R e. _V )
20		isps 17202	. . 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 17	. 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 1245	1 |- ( R e. PosetRel -> `' R e. PosetRel )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ps 17200

This theorem is referenced by:  cnvpsb  17213  cnvtsr  17222  ordtcnv  21005  xrge0iifhmeo  29982