Theorem oduval 17130

Description: Value of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)

Hypotheses

Ref	Expression
oduval.d	|- D = (ODual ` O )
oduval.l	|- .<_ = ( le ` O )

Assertion

Ref	Expression
oduval	|- D = ( O sSet <. ( le ` ndx ) , `' .<_ >. )

Proof of Theorem oduval

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 |- ( a = O -> a = O )
2		fveq2 6191	. . . . . . 7 |- ( a = O -> ( le ` a ) = ( le ` O ) )
3	2	cnveqd 5298	. . . . . 6 |- ( a = O -> `' ( le ` a ) = `' ( le ` O ) )
4	3	opeq2d 4409	. . . . 5 |- ( a = O -> <. ( le ` ndx ) , `' ( le ` a ) >. = <. ( le ` ndx ) , `' ( le ` O ) >. )
5	1, 4	oveq12d 6668	. . . 4 |- ( a = O -> ( a sSet <. ( le ` ndx ) , `' ( le ` a ) >. ) = ( O sSet <. ( le ` ndx ) , `' ( le ` O ) >. ) )
6		df-odu 17129	. . . 4 |- ODual = ( a e. _V |-> ( a sSet <. ( le ` ndx ) , `' ( le ` a ) >. ) )
7		ovex 6678	. . . 4 |- ( O sSet <. ( le ` ndx ) , `' ( le ` O ) >. ) e. _V
8	5, 6, 7	fvmpt 6282	. . 3 |- ( O e. _V -> (ODual ` O ) = ( O sSet <. ( le ` ndx ) , `' ( le ` O ) >. ) )
9		fvprc 6185	. . . 4 |- ( -. O e. _V -> (ODual ` O ) = (/) )
10		reldmsets 15886	. . . . 5 |- Rel dom sSet
11	10	ovprc1 6684	. . . 4 |- ( -. O e. _V -> ( O sSet <. ( le ` ndx ) , `' ( le ` O ) >. ) = (/) )
12	9, 11	eqtr4d 2659	. . 3 |- ( -. O e. _V -> (ODual ` O ) = ( O sSet <. ( le ` ndx ) , `' ( le ` O ) >. ) )
13	8, 12	pm2.61i 176	. 2 |- (ODual ` O ) = ( O sSet <. ( le ` ndx ) , `' ( le ` O ) >. )
14		oduval.d	. 2 |- D = (ODual ` O )
15		oduval.l	. . . . 5 |- .<_ = ( le ` O )
16	15	cnveqi 5297	. . . 4 |- `' .<_ = `' ( le ` O )
17	16	opeq2i 4406	. . 3 |- <. ( le ` ndx ) , `' .<_ >. = <. ( le ` ndx ) , `' ( le ` O ) >.
18	17	oveq2i 6661	. 2 |- ( O sSet <. ( le ` ndx ) , `' .<_ >. ) = ( O sSet <. ( le ` ndx ) , `' ( le ` O ) >. )
19	13, 14, 18	3eqtr4i 2654	1 |- D = ( O sSet <. ( le ` ndx ) , `' .<_ >. )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   <.cop 4183   `'ccnv 5113   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855   lecple 15948  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

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-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sets 15864  df-odu 17129

This theorem is referenced by:  oduleval  17131  odubas  17133