Theorem cnvps 16041

Description: The converse of a poset is a poset. In the general case ( `' R e. PosetRel -> R e. PosetRel ) is not true. See cnvpsb 16042 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 5362	. . 3 |- Rel `' R
2	1	a1i 11	. 2 |- ( R e. PosetRel -> Rel `' R )
3		cnvco 5177	. . 3 |- `' ( R o. R ) = ( `' R o. `' R )
4		pstr2 16034	. . . 4 |- ( R e. PosetRel -> ( R o. R ) C_ R )
5		cnvss 5164	. . . 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 3534	. 2 |- ( R e. PosetRel -> ( `' R o. `' R ) C_ `' R )
8		psrel 16032	. . . . . 6 |- ( R e. PosetRel -> Rel R )
9		dfrel2 5441	. . . . . 6 |- ( Rel R <-> `' `' R = R )
10	8, 9	sylib 196	. . . . 5 |- ( R e. PosetRel -> `' `' R = R )
11	10	ineq2d 3686	. . . 4 |- ( R e. PosetRel -> ( `' R i^i `' `' R ) = ( `' R i^i R ) )
12		incom 3677	. . . 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 16033	. . 3 |- ( R e. PosetRel -> ( R i^i `' R ) = ( _I |` U. U. R ) )
15		relcnvfld 5521	. . . . 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 5262	. . 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 6719	. . 3 |- ( R e. PosetRel -> `' R e. _V )
20		isps 16031	. . 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 1177	1 |- ( R e. PosetRel -> `' R e. PosetRel )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 971   = wceq 1398   e. wcel 1823   _Vcvv 3106   i^i cin 3460   C_ wss 3461   U.cuni 4235   _I cid 4779   `'ccnv 4987   |` cres 4990   o. ccom 4992   Rel wrel 4993   PosetRelcps 16027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ps 16029

This theorem is referenced by:  cnvpsb  16042  cnvtsr  16051  ordtcnv  19869  xrge0iifhmeo  28153