Theorem odupos 17135

Description: Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)

Hypothesis

Ref	Expression
odupos.d	|- D = (ODual ` O )

Assertion

Ref	Expression
odupos	|- ( O e. Poset -> D e. Poset )

Proof of Theorem odupos

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		odupos.d	. . . 4 |- D = (ODual ` O )
2		fvex 6201	. . . 4 |- (ODual ` O ) e. _V
3	1, 2	eqeltri 2697	. . 3 |- D e. _V
4	3	a1i 11	. 2 |- ( O e. Poset -> D e. _V )
5		eqid 2622	. . . 4 |- ( Base ` O ) = ( Base ` O )
6	1, 5	odubas 17133	. . 3 |- ( Base ` O ) = ( Base ` D )
7	6	a1i 11	. 2 |- ( O e. Poset -> ( Base ` O ) = ( Base ` D ) )
8		eqid 2622	. . . 4 |- ( le ` O ) = ( le ` O )
9	1, 8	oduleval 17131	. . 3 |- `' ( le ` O ) = ( le ` D )
10	9	a1i 11	. 2 |- ( O e. Poset -> `' ( le ` O ) = ( le ` D ) )
11	5, 8	posref 16951	. . 3 |- ( ( O e. Poset /\ a e. ( Base ` O ) ) -> a ( le ` O ) a )
12		vex 3203	. . . 4 |- a e. _V
13	12, 12	brcnv 5305	. . 3 |- ( a `' ( le ` O ) a <-> a ( le ` O ) a )
14	11, 13	sylibr 224	. 2 |- ( ( O e. Poset /\ a e. ( Base ` O ) ) -> a `' ( le ` O ) a )
15		vex 3203	. . . . 5 |- b e. _V
16	12, 15	brcnv 5305	. . . 4 |- ( a `' ( le ` O ) b <-> b ( le ` O ) a )
17	15, 12	brcnv 5305	. . . 4 |- ( b `' ( le ` O ) a <-> a ( le ` O ) b )
18	16, 17	anbi12ci 734	. . 3 |- ( ( a `' ( le ` O ) b /\ b `' ( le ` O ) a ) <-> ( a ( le ` O ) b /\ b ( le ` O ) a ) )
19	5, 8	posasymb 16952	. . . 4 |- ( ( O e. Poset /\ a e. ( Base ` O ) /\ b e. ( Base ` O ) ) -> ( ( a ( le ` O ) b /\ b ( le ` O ) a ) <-> a = b ) )
20	19	biimpd 219	. . 3 |- ( ( O e. Poset /\ a e. ( Base ` O ) /\ b e. ( Base ` O ) ) -> ( ( a ( le ` O ) b /\ b ( le ` O ) a ) -> a = b ) )
21	18, 20	syl5bi 232	. 2 |- ( ( O e. Poset /\ a e. ( Base ` O ) /\ b e. ( Base ` O ) ) -> ( ( a `' ( le ` O ) b /\ b `' ( le ` O ) a ) -> a = b ) )
22		3anrev 1049	. . . 4 |- ( ( a e. ( Base ` O ) /\ b e. ( Base ` O ) /\ c e. ( Base ` O ) ) <-> ( c e. ( Base ` O ) /\ b e. ( Base ` O ) /\ a e. ( Base ` O ) ) )
23	5, 8	postr 16953	. . . 4 |- ( ( O e. Poset /\ ( c e. ( Base ` O ) /\ b e. ( Base ` O ) /\ a e. ( Base ` O ) ) ) -> ( ( c ( le ` O ) b /\ b ( le ` O ) a ) -> c ( le ` O ) a ) )
24	22, 23	sylan2b 492	. . 3 |- ( ( O e. Poset /\ ( a e. ( Base ` O ) /\ b e. ( Base ` O ) /\ c e. ( Base ` O ) ) ) -> ( ( c ( le ` O ) b /\ b ( le ` O ) a ) -> c ( le ` O ) a ) )
25		vex 3203	. . . . 5 |- c e. _V
26	15, 25	brcnv 5305	. . . 4 |- ( b `' ( le ` O ) c <-> c ( le ` O ) b )
27	16, 26	anbi12ci 734	. . 3 |- ( ( a `' ( le ` O ) b /\ b `' ( le ` O ) c ) <-> ( c ( le ` O ) b /\ b ( le ` O ) a ) )
28	12, 25	brcnv 5305	. . 3 |- ( a `' ( le ` O ) c <-> c ( le ` O ) a )
29	24, 27, 28	3imtr4g 285	. 2 |- ( ( O e. Poset /\ ( a e. ( Base ` O ) /\ b e. ( Base ` O ) /\ c e. ( Base ` O ) ) ) -> ( ( a `' ( le ` O ) b /\ b `' ( le ` O ) c ) -> a `' ( le ` O ) c ) )
30	4, 7, 10, 14, 21, 29	isposd 16955	1 |- ( O e. Poset -> D e. Poset )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113   ` cfv 5888   Basecbs 15857   lecple 15948   Posetcpo 16940  ODualcodu 17128

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-dec 11494  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ple 15961  df-preset 16928  df-poset 16946  df-odu 17129

This theorem is referenced by:  oduposb  17136  posglbd  17150  odutos  29663