Theorem cnvpsb 17213

Description: The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)

Assertion

Ref	Expression
cnvpsb	|- ( Rel R -> ( R e. PosetRel <-> `' R e. PosetRel ) )

Proof of Theorem cnvpsb

Step	Hyp	Ref	Expression
1		cnvps 17212	. 2 |- ( R e. PosetRel -> `' R e. PosetRel )
2		cnvps 17212	. . 3 |- ( `' R e. PosetRel -> `' `' R e. PosetRel )
3		dfrel2 5583	. . . 4 |- ( Rel R <-> `' `' R = R )
4		eleq1 2689	. . . . 5 |- ( `' `' R = R -> ( `' `' R e. PosetRel <-> R e. PosetRel ) )
5	4	biimpd 219	. . . 4 |- ( `' `' R = R -> ( `' `' R e. PosetRel -> R e. PosetRel ) )
6	3, 5	sylbi 207	. . 3 |- ( Rel R -> ( `' `' R e. PosetRel -> R e. PosetRel ) )
7	2, 6	syl5 34	. 2 |- ( Rel R -> ( `' R e. PosetRel -> R e. PosetRel ) )
8	1, 7	impbid2 216	1 |- ( Rel R -> ( R e. PosetRel <-> `' R e. PosetRel ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   `'ccnv 5113   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: (None)