Theorem ordtprsuni 29293

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 ∈ Preset → 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 29288	. . . . 5 ⊢ (𝐾 ∈ Preset → dom ≤ = 𝐵)
4	3	sneqd 4137	. . . 4 ⊢ (𝐾 ∈ Preset → {dom ≤ } = {𝐵})
5		biidd 251	. . . . . . . 8 ⊢ (𝐾 ∈ Preset → (¬ 𝑦 ≤ 𝑥 ↔ ¬ 𝑦 ≤ 𝑥))
6	3, 5	rabeqbidv 3168	. . . . . . 7 ⊢ (𝐾 ∈ Preset → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
7	3, 6	mpteq12dv 4663	. . . . . 6 ⊢ (𝐾 ∈ Preset → (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
8	7	rneqd 5274	. . . . 5 ⊢ (𝐾 ∈ Preset → ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
9		biidd 251	. . . . . . . 8 ⊢ (𝐾 ∈ Preset → (¬ 𝑥 ≤ 𝑦 ↔ ¬ 𝑥 ≤ 𝑦))
10	3, 9	rabeqbidv 3168	. . . . . . 7 ⊢ (𝐾 ∈ Preset → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
11	3, 10	mpteq12dv 4663	. . . . . 6 ⊢ (𝐾 ∈ Preset → (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
12	11	rneqd 5274	. . . . 5 ⊢ (𝐾 ∈ Preset → ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
13	8, 12	uneq12d 3730	. . . 4 ⊢ (𝐾 ∈ Preset → (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 3730	. . 3 ⊢ (𝐾 ∈ Preset → ({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 4382	. 2 ⊢ (𝐾 ∈ Preset → ∪ ({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 6113	. . . . . 6 ⊢ (le‘𝐾) ∈ V
17	16	inex1 4727	. . . . 5 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
18	2, 17	eqeltri 2684	. . . 4 ⊢ ≤ ∈ V
19		eqid 2610	. . . . 5 ⊢ dom ≤ = dom ≤
20		eqid 2610	. . . . 5 ⊢ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥})
21		eqid 2610	. . . . 5 ⊢ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦})
22	19, 20, 21	ordtuni 20804	. . . 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	23, 3	syl5reqr 2659	. 2 ⊢ (𝐾 ∈ Preset → 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 3727	. . . . 5 ⊢ (𝐸 ∪ 𝐹) = (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
28	27	a1i 11	. . . 4 ⊢ (𝐾 ∈ Preset → (𝐸 ∪ 𝐹) = (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})))
29	28	uneq2d 3729	. . 3 ⊢ (𝐾 ∈ Preset → ({𝐵} ∪ (𝐸 ∪ 𝐹)) = ({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))))
30	29	unieqd 4382	. 2 ⊢ (𝐾 ∈ Preset → ∪ ({𝐵} ∪ (𝐸 ∪ 𝐹)) = ∪ ({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))))
31	15, 24, 30	3eqtr4d 2654	1 ⊢ (𝐾 ∈ Preset → 𝐵 = ∪ ({𝐵} ∪ (𝐸 ∪ 𝐹)))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∪ cun 3538   ∩ cin 3539  {csn 4125  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039  ‘cfv 5804  Basecbs 15695  lecple 15775   Preset cpreset 16749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-preset 16751

This theorem is referenced by:  ordtrest2NEW  29297