Theorem ordtprsuni 34254

Description: Value of the order topology. (Contributed by Thierry Arnoux, 13-Sep-2018.)

Hypotheses

Ref	Expression
ordtNEW.b	⊢ 𝐵 = (Base‘𝐾)
ordtNEW.l	⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
ordtposval.e	⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
ordtposval.f	⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})

Assertion

Ref	Expression
ordtprsuni	⊢ (𝐾 ∈ Proset → 𝐵 = ∪ ({𝐵} ∪ (𝐸 ∪ 𝐹)))

Distinct variable groups:   𝑥,𝑦, ≤   𝑥,𝐵,𝑦   𝑥,𝐾,𝑦

Allowed substitution hints:   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem ordtprsuni

Step	Hyp	Ref	Expression
1		ordtNEW.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
2		ordtNEW.l	. . . . . 6 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
3	1, 2	prsdm 34249	. . . . 5 ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵)
4	3	sneqd 4606	. . . 4 ⊢ (𝐾 ∈ Proset → {dom ≤ } = {𝐵})
5		biidd 265	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (¬ 𝑦 ≤ 𝑥 ↔ ¬ 𝑦 ≤ 𝑥))
6	3, 5	rabeqbidv 3441	. . . . . . 7 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
7	3, 6	mpteq12dv 5202	. . . . . 6 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
8	7	rneqd 5929	. . . . 5 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
9		biidd 265	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (¬ 𝑥 ≤ 𝑦 ↔ ¬ 𝑥 ≤ 𝑦))
10	3, 9	rabeqbidv 3441	. . . . . . 7 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
11	3, 10	mpteq12dv 5202	. . . . . 6 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
12	11	rneqd 5929	. . . . 5 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
13	8, 12	uneq12d 4131	. . . 4 ⊢ (𝐾 ∈ Proset → (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦})) = (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})))
14	4, 13	uneq12d 4131	. . 3 ⊢ (𝐾 ∈ Proset → ({dom ≤ } ∪ (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}))) = ({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))))
15	14	unieqd 4889	. 2 ⊢ (𝐾 ∈ Proset → ∪ ({dom ≤ } ∪ (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}))) = ∪ ({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))))
16		fvex 6895	. . . . . 6 ⊢ (le‘𝐾) ∈ V
17	16	inex1 5288	. . . . 5 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
18	2, 17	eqeltri 2865	. . . 4 ⊢ ≤ ∈ V
19		eqid 2769	. . . . 5 ⊢ dom ≤ = dom ≤
20		eqid 2769	. . . . 5 ⊢ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥})
21		eqid 2769	. . . . 5 ⊢ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦})
22	19, 20, 21	ordtuni 23316	. . . 4 ⊢ ( ≤ ∈ V → dom ≤ = ∪ ({dom ≤ } ∪ (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}))))
23	18, 22	ax-mp 5	. . 3 ⊢ dom ≤ = ∪ ({dom ≤ } ∪ (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦})))
24	3, 23	eqtr3di 2819	. 2 ⊢ (𝐾 ∈ Proset → 𝐵 = ∪ ({dom ≤ } ∪ (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}))))
25		ordtposval.e	. . . . . 6 ⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
26		ordtposval.f	. . . . . 6 ⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
27	25, 26	uneq12i 4128	. . . . 5 ⊢ (𝐸 ∪ 𝐹) = (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
28	27	a1i 11	. . . 4 ⊢ (𝐾 ∈ Proset → (𝐸 ∪ 𝐹) = (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})))
29	28	uneq2d 4130	. . 3 ⊢ (𝐾 ∈ Proset → ({𝐵} ∪ (𝐸 ∪ 𝐹)) = ({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))))
30	29	unieqd 4889	. 2 ⊢ (𝐾 ∈ Proset → ∪ ({𝐵} ∪ (𝐸 ∪ 𝐹)) = ∪ ({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))))
31	15, 24, 30	3eqtr4d 2814	1 ⊢ (𝐾 ∈ Proset → 𝐵 = ∪ ({𝐵} ∪ (𝐸 ∪ 𝐹)))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1567   ∈ wcel 2149  {crab 3423  Vcvv 3463   ∪ cun 3911   ∩ cin 3912  {csn 4594  ∪ cuni 4876   class class class wbr 5113   ↦ cmpt 5196   × cxp 5660  dom cdm 5662  ran crn 5663  ‘cfv 6537  Basecbs 17269  lecple 17317   Proset cproset 18348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-proset 18350

This theorem is referenced by:  ordtrest2NEW  34258