Theorem ordtcnv 23427

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 2762	. . . . . . . 8 ⊢ dom 𝑅 = dom 𝑅
2	1	psrn 18667	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
3	2	eqcomd 2768	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 = dom 𝑅)
4	3	sneqd 4599	. . . . 5 ⊢ (𝑅 ∈ PosetRel → {ran 𝑅} = {dom 𝑅})
5		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 5866	. . . . . . . . . . . 12 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (𝑅 ∈ PosetRel → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
9	8	notbid 321	. . . . . . . . . 10 ⊢ (𝑅 ∈ PosetRel → (¬ 𝑦◡𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦))
10	3, 9	rabeqbidv 3432	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥} = {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦})
11	3, 10	mpteq12dv 5196	. . . . . . . 8 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))
12	11	rneqd 5926	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))
13	6, 5	brcnv 5866	. . . . . . . . . . . 12 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝑅 ∈ PosetRel → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
15	14	notbid 321	. . . . . . . . . 10 ⊢ (𝑅 ∈ PosetRel → (¬ 𝑥◡𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
16	3, 15	rabeqbidv 3432	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦} = {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥})
17	3, 16	mpteq12dv 5196	. . . . . . . 8 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}) = (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
18	17	rneqd 5926	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
19	12, 18	uneq12d 4119	. . . . . 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 4108	. . . . . 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 2813	. . . . 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 4119	. . . 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 6886	. . 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 6886	. 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 18670	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)
26		df-rn 5670	. . . 4 ⊢ ran 𝑅 = dom ◡𝑅
27		eqid 2762	. . . 4 ⊢ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})
28		eqid 2762	. . . 4 ⊢ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥◡𝑅𝑦})
29	26, 27, 28	ordtval 23415	. . 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 18	. 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 2762	. . 3 ⊢ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥})
32		eqid 2762	. . 3 ⊢ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦})
33	1, 31, 32	ordtval 23415	. 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 2807	1 ⊢ (𝑅 ∈ PosetRel → (ordTop‘◡𝑅) = (ordTop‘𝑅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {crab 3414   ∪ cun 3900  {csn 4587   class class class wbr 5107   ↦ cmpt 5190  ^◡ccnv 5658  dom cdm 5659  ran crn 5660  ‘cfv 6537  ficfi 9383  topGenctg 17526  ordTopcordt 17589  PosetRelcps 18656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6493  df-fun 6539  df-fv 6545  df-ordt 17591  df-ps 18658

This theorem is used by:  ordtrest2  23430  cnvordtrestixx  34410