Theorem cnvps 15504

Description: The converse of a poset is a poset. In the general case ( `' R e. PosetRel -> R e. PosetRel ) is not true. See cnvpsb 15505 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 5317	. . 3 |- Rel `' R
2	1	a1i 11	. 2 |- ( R e. PosetRel -> Rel `' R )
3		cnvco 5136	. . 3 |- `' ( R o. R ) = ( `' R o. `' R )
4		pstr2 15497	. . . 4 |- ( R e. PosetRel -> ( R o. R ) C_ R )
5		cnvss 5123	. . . 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 3512	. 2 |- ( R e. PosetRel -> ( `' R o. `' R ) C_ `' R )
8		psrel 15495	. . . . . 6 |- ( R e. PosetRel -> Rel R )
9		dfrel2 5399	. . . . . 6 |- ( Rel R <-> `' `' R = R )
10	8, 9	sylib 196	. . . . 5 |- ( R e. PosetRel -> `' `' R = R )
11	10	ineq2d 3663	. . . 4 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( `' R i^i R ) )
12		incom 3654	. . . 4 |- ( `' R i^i R ) = ( R i^i `' R )
13	11, 12	syl6eq 2511	. . 3 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( R i^i `' R ) )
14		psref2 15496	. . 3 |- ( R e. PosetRel -> ( R i^i `' R ) = ( _I |` U. U. R ) )
15		relcnvfld 5479	. . . . 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 5221	. . 3 |- ( R e. PosetRel -> ( _I |` U. U. R ) = ( _I |` U. U. `' R ) )
18	13, 14, 17	3eqtrd 2499	. 2 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( _I |` U. U. `' R ) )
19		cnvexg 6637	. . 3 |- ( R e. PosetRel -> `' R e. _V )
20		isps 15494	. . 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 1370   e. wcel 1758   _Vcvv 3078   i^i cin 3438   C_ wss 3439   U.cuni 4202   _I cid 4742   `'ccnv 4950   |` cres 4953   o. ccom 4955   Rel wrel 4956   PosetRelcps 15490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ps 15492

This theorem is referenced by:  cnvpsb  15505  cnvtsr  15514  ordtcnv  18940  xrge0iifhmeo  26531