Theorem ordtcnv 23116

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 2731	. . . . . . . 8 ⊢ dom 𝑅 = dom 𝑅
2	1	psrn 18481	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
3	2	eqcomd 2737	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 = dom 𝑅)
4	3	sneqd 4585	. . . . 5 ⊢ (𝑅 ∈ PosetRel → {ran 𝑅} = {dom 𝑅})
5		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 5821	. . . . . . . . . . . 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 3413	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥} = {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦})
11	3, 10	mpteq12dv 5176	. . . . . . . 8 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))
12	11	rneqd 5877	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))
13	6, 5	brcnv 5821	. . . . . . . . . . . 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 3413	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦} = {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥})
17	3, 16	mpteq12dv 5176	. . . . . . . 8 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}) = (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
18	17	rneqd 5877	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
19	12, 18	uneq12d 4116	. . . . . 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 4105	. . . . . 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 2782	. . . . 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 4116	. . . 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 6826	. . 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 6826	. 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 18484	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)
26		df-rn 5625	. . . 4 ⊢ ran 𝑅 = dom ◡𝑅
27		eqid 2731	. . . 4 ⊢ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})
28		eqid 2731	. . . 4 ⊢ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦})
29	26, 27, 28	ordtval 23104	. . 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 2731	. . 3 ⊢ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥})
32		eqid 2731	. . 3 ⊢ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦})
33	1, 31, 32	ordtval 23104	. 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 2776	1 ⊢ (𝑅 ∈ PosetRel → (ordTop‘◡𝑅) = (ordTop‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  {crab 3395   ∪ cun 3895  {csn 4573   class class class wbr 5089   ↦ cmpt 5170  ^◡ccnv 5613  dom cdm 5614  ran crn 5615  ‘cfv 6481  ficfi 9294  topGenctg 17341  ordTopcordt 17403  PosetRelcps 18470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-ordt 17405  df-ps 18472

This theorem is referenced by:  ordtrest2  23119  cnvordtrestixx  33926