Theorem ordtprsuni 34078

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 34073	. . . . 5 ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵)
4	3	sneqd 4592	. . . 4 ⊢ (𝐾 ∈ Proset → {dom ≤ } = {𝐵})
5		biidd 262	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (¬ 𝑦 ≤ 𝑥 ↔ ¬ 𝑦 ≤ 𝑥))
6	3, 5	rabeqbidv 3417	. . . . . . 7 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
7	3, 6	mpteq12dv 5185	. . . . . 6 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
8	7	rneqd 5887	. . . . 5 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
9		biidd 262	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (¬ 𝑥 ≤ 𝑦 ↔ ¬ 𝑥 ≤ 𝑦))
10	3, 9	rabeqbidv 3417	. . . . . . 7 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
11	3, 10	mpteq12dv 5185	. . . . . 6 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
12	11	rneqd 5887	. . . . 5 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
13	8, 12	uneq12d 4121	. . . 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 4121	. . 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 4876	. 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 6847	. . . . . 6 ⊢ (le‘𝐾) ∈ V
17	16	inex1 5262	. . . . 5 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
18	2, 17	eqeltri 2832	. . . 4 ⊢ ≤ ∈ V
19		eqid 2736	. . . . 5 ⊢ dom ≤ = dom ≤
20		eqid 2736	. . . . 5 ⊢ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥})
21		eqid 2736	. . . . 5 ⊢ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦})
22	19, 20, 21	ordtuni 23136	. . . 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 2786	. 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 4118	. . . . 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 4120	. . 3 ⊢ (𝐾 ∈ Proset → ({𝐵} ∪ (𝐸 ∪ 𝐹)) = ({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))))
30	29	unieqd 4876	. 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 2781	1 ⊢ (𝐾 ∈ Proset → 𝐵 = ∪ ({𝐵} ∪ (𝐸 ∪ 𝐹)))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   ∪ cun 3899   ∩ cin 3900  {csn 4580  ∪ cuni 4863   class class class wbr 5098   ↦ cmpt 5179   × cxp 5622  dom cdm 5624  ran crn 5625  ‘cfv 6492  Basecbs 17138  lecple 17186   Proset cproset 18217

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-proset 18219

This theorem is referenced by:  ordtrest2NEW  34082