Theorem ordtcnv 21005

Description: The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
ordtcnv	|- ( R e. PosetRel -> (ordTop ` `' R ) = (ordTop ` R ) )

Proof of Theorem ordtcnv

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . 8 |- dom R = dom R
2	1	psrn 17209	. . . . . . 7 |- ( R e. PosetRel -> dom R = ran R )
3	2	eqcomd 2628	. . . . . 6 |- ( R e. PosetRel -> ran R = dom R )
4	3	sneqd 4189	. . . . 5 |- ( R e. PosetRel -> { ran R } = { dom R } )
5		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
6		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
7	5, 6	brcnv 5305	. . . . . . . . . . . 12 |- ( y `' R x <-> x R y )
8	7	a1i 11	. . . . . . . . . . 11 |- ( R e. PosetRel -> ( y `' R x <-> x R y ) )
9	8	notbid 308	. . . . . . . . . 10 |- ( R e. PosetRel -> ( -. y `' R x <-> -. x R y ) )
10	3, 9	rabeqbidv 3195	. . . . . . . . 9 |- ( R e. PosetRel -> { y e. ran R | -. y `' R x } = { y e. dom R | -. x R y } )
11	3, 10	mpteq12dv 4733	. . . . . . . 8 |- ( R e. PosetRel -> ( x e. ran R |-> { y e. ran R | -. y `' R x } ) = ( x e. dom R |-> { y e. dom R | -. x R y } ) )
12	11	rneqd 5353	. . . . . . 7 |- ( R e. PosetRel -> ran ( x e. ran R |-> { y e. ran R | -. y `' R x } ) = ran ( x e. dom R |-> { y e. dom R | -. x R y } ) )
13	6, 5	brcnv 5305	. . . . . . . . . . . 12 |- ( x `' R y <-> y R x )
14	13	a1i 11	. . . . . . . . . . 11 |- ( R e. PosetRel -> ( x `' R y <-> y R x ) )
15	14	notbid 308	. . . . . . . . . 10 |- ( R e. PosetRel -> ( -. x `' R y <-> -. y R x ) )
16	3, 15	rabeqbidv 3195	. . . . . . . . 9 |- ( R e. PosetRel -> { y e. ran R | -. x `' R y } = { y e. dom R | -. y R x } )
17	3, 16	mpteq12dv 4733	. . . . . . . 8 |- ( R e. PosetRel -> ( x e. ran R |-> { y e. ran R | -. x `' R y } ) = ( x e. dom R |-> { y e. dom R | -. y R x } ) )
18	17	rneqd 5353	. . . . . . 7 |- ( R e. PosetRel -> ran ( x e. ran R |-> { y e. ran R | -. x `' R y } ) = ran ( x e. dom R |-> { y e. dom R | -. y R x } ) )
19	12, 18	uneq12d 3768	. . . . . 6 |- ( R e. PosetRel -> ( ran ( x e. ran R |-> { y e. ran R | -. y `' R x } ) u. ran ( x e. ran R |-> { y e. ran R | -. x `' R y } ) ) = ( ran ( x e. dom R |-> { y e. dom R | -. x R y } ) u. ran ( x e. dom R |-> { y e. dom R | -. y R x } ) ) )
20		uncom 3757	. . . . . 6 |- ( ran ( x e. dom R |-> { y e. dom R | -. x R y } ) u. ran ( x e. dom R |-> { y e. dom R | -. y R x } ) ) = ( ran ( x e. dom R |-> { y e. dom R | -. y R x } ) u. ran ( x e. dom R |-> { y e. dom R | -. x R y } ) )
21	19, 20	syl6eq 2672	. . . . 5 |- ( R e. PosetRel -> ( ran ( x e. ran R |-> { y e. ran R | -. y `' R x } ) u. ran ( x e. ran R |-> { y e. ran R | -. x `' R y } ) ) = ( ran ( x e. dom R |-> { y e. dom R | -. y R x } ) u. ran ( x e. dom R |-> { y e. dom R | -. x R y } ) ) )
22	4, 21	uneq12d 3768	. . . 4 |- ( R e. PosetRel -> ( { ran R } u. ( ran ( x e. ran R |-> { y e. ran R | -. y `' R x } ) u. ran ( x e. ran R |-> { y e. ran R | -. x `' R y } ) ) ) = ( { dom R } u. ( ran ( x e. dom R |-> { y e. dom R | -. y R x } ) u. ran ( x e. dom R |-> { y e. dom R | -. x R y } ) ) ) )
23	22	fveq2d 6195	. . 3 |- ( R e. PosetRel -> ( fi ` ( { ran R } u. ( ran ( x e. ran R |-> { y e. ran R | -. y `' R x } ) u. ran ( x e. ran R |-> { y e. ran R | -. x `' R y } ) ) ) ) = ( fi ` ( { dom R } u. ( ran ( x e. dom R |-> { y e. dom R | -. y R x } ) u. ran ( x e. dom R |-> { y e. dom R | -. x R y } ) ) ) ) )
24	23	fveq2d 6195	. 2 |- ( R e. PosetRel -> ( topGen ` ( fi ` ( { ran R } u. ( ran ( x e. ran R |-> { y e. ran R | -. y `' R x } ) u. ran ( x e. ran R |-> { y e. ran R | -. x `' R y } ) ) ) ) ) = ( topGen ` ( fi ` ( { dom R } u. ( ran ( x e. dom R |-> { y e. dom R | -. y R x } ) u. ran ( x e. dom R |-> { y e. dom R | -. x R y } ) ) ) ) ) )
25		cnvps 17212	. . 3 |- ( R e. PosetRel -> `' R e. PosetRel )
26		df-rn 5125	. . . 4 |- ran R = dom `' R
27		eqid 2622	. . . 4 |- ran ( x e. ran R |-> { y e. ran R | -. y `' R x } ) = ran ( x e. ran R |-> { y e. ran R | -. y `' R x } )
28		eqid 2622	. . . 4 |- ran ( x e. ran R |-> { y e. ran R | -. x `' R y } ) = ran ( x e. ran R |-> { y e. ran R | -. x `' R y } )
29	26, 27, 28	ordtval 20993	. . 3 |- ( `' R e. PosetRel -> (ordTop ` `' R ) = ( topGen ` ( fi ` ( { ran R } u. ( ran ( x e. ran R |-> { y e. ran R | -. y `' R x } ) u. ran ( x e. ran R |-> { y e. ran R | -. x `' R y } ) ) ) ) ) )
30	25, 29	syl 17	. 2 |- ( R e. PosetRel -> (ordTop ` `' R ) = ( topGen ` ( fi ` ( { ran R } u. ( ran ( x e. ran R |-> { y e. ran R | -. y `' R x } ) u. ran ( x e. ran R |-> { y e. ran R | -. x `' R y } ) ) ) ) ) )
31		eqid 2622	. . 3 |- ran ( x e. dom R |-> { y e. dom R | -. y R x } ) = ran ( x e. dom R |-> { y e. dom R | -. y R x } )
32		eqid 2622	. . 3 |- ran ( x e. dom R |-> { y e. dom R | -. x R y } ) = ran ( x e. dom R |-> { y e. dom R | -. x R y } )
33	1, 31, 32	ordtval 20993	. 2 |- ( R e. PosetRel -> (ordTop ` R ) = ( topGen ` ( fi ` ( { dom R } u. ( ran ( x e. dom R |-> { y e. dom R | -. y R x } ) u. ran ( x e. dom R |-> { y e. dom R | -. x R y } ) ) ) ) ) )
34	24, 30, 33	3eqtr4d 2666	1 |- ( R e. PosetRel -> (ordTop ` `' R ) = (ordTop ` R ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {crab 2916   u. cun 3572   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   ` cfv 5888   ficfi 8316   topGenctg 16098  ordTopcordt 16159   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-sbc 3436  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ordt 16161  df-ps 17200

This theorem is referenced by:  ordtrest2  21008  cnvordtrestixx  29959