Theorem ordtcnv 23209

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

Assertion

Ref	Expression
ordtcnv	⊢ (𝑅 ∈ 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 2737	. . . . . . . 8 ⊢ dom 𝑅 = dom 𝑅
2	1	psrn 18620	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
3	2	eqcomd 2743	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 = dom 𝑅)
4	3	sneqd 4638	. . . . 5 ⊢ (𝑅 ∈ PosetRel → {ran 𝑅} = {dom 𝑅})
5		vex 3484	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6		vex 3484	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 5893	. . . . . . . . . . . 12 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (𝑅 ∈ PosetRel → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
9	8	notbid 318	. . . . . . . . . 10 ⊢ (𝑅 ∈ PosetRel → (¬ 𝑦◡𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦))
10	3, 9	rabeqbidv 3455	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥} = {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦})
11	3, 10	mpteq12dv 5233	. . . . . . . 8 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))
12	11	rneqd 5949	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))
13	6, 5	brcnv 5893	. . . . . . . . . . . 12 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝑅 ∈ PosetRel → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
15	14	notbid 318	. . . . . . . . . 10 ⊢ (𝑅 ∈ PosetRel → (¬ 𝑥◡𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
16	3, 15	rabeqbidv 3455	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦} = {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥})
17	3, 16	mpteq12dv 5233	. . . . . . . 8 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}) = (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
18	17	rneqd 5949	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
19	12, 18	uneq12d 4169	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦})) = (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥})))
20		uncom 4158	. . . . . 6 ⊢ (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥})) = (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))
21	19, 20	eqtrdi 2793	. . . . 5 ⊢ (𝑅 ∈ PosetRel → (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦})) = (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦})))
22	4, 21	uneq12d 4169	. . . 4 ⊢ (𝑅 ∈ PosetRel → ({ran 𝑅} ∪ (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}))) = ({dom 𝑅} ∪ (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))))
23	22	fveq2d 6910	. . 3 ⊢ (𝑅 ∈ PosetRel → (fi‘({ran 𝑅} ∪ (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦})))) = (fi‘({dom 𝑅} ∪ (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦})))))
24	23	fveq2d 6910	. 2 ⊢ (𝑅 ∈ PosetRel → (topGen‘(fi‘({ran 𝑅} ∪ (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}))))) = (topGen‘(fi‘({dom 𝑅} ∪ (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))))))
25		cnvps 18623	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)
26		df-rn 5696	. . . 4 ⊢ ran 𝑅 = dom ◡𝑅
27		eqid 2737	. . . 4 ⊢ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})
28		eqid 2737	. . . 4 ⊢ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦})
29	26, 27, 28	ordtval 23197	. . 3 ⊢ (◡𝑅 ∈ PosetRel → (ordTop‘◡𝑅) = (topGen‘(fi‘({ran 𝑅} ∪ (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}))))))
30	25, 29	syl 17	. 2 ⊢ (𝑅 ∈ PosetRel → (ordTop‘◡𝑅) = (topGen‘(fi‘({ran 𝑅} ∪ (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}))))))
31		eqid 2737	. . 3 ⊢ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥})
32		eqid 2737	. . 3 ⊢ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦})
33	1, 31, 32	ordtval 23197	. 2 ⊢ (𝑅 ∈ PosetRel → (ordTop‘𝑅) = (topGen‘(fi‘({dom 𝑅} ∪ (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))))))
34	24, 30, 33	3eqtr4d 2787	1 ⊢ (𝑅 ∈ PosetRel → (ordTop‘◡𝑅) = (ordTop‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  {crab 3436   ∪ cun 3949  {csn 4626   class class class wbr 5143   ↦ cmpt 5225  ^◡ccnv 5684  dom cdm 5685  ran crn 5686  ‘cfv 6561  ficfi 9450  topGenctg 17482  ordTopcordt 17544  PosetRelcps 18609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ordt 17546  df-ps 18611

This theorem is referenced by:  ordtrest2  23212  cnvordtrestixx  33912