Theorem ordtprsval 34597

Description: Value of the order topology for a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)

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
ordtprsval	⊢ (𝐾 ∈ Proset → (ordTop‘ ≤ ) = (topGen‘(fi‘({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 ordtprsval

Step	Hyp	Ref	Expression
1		ordtNEW.l	. . . 4 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
2		fvex 6899	. . . . 5 ⊢ (le‘𝐾) ∈ V
3	2	inex1 5276	. . . 4 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
4	1, 3	eqeltri 2856	. . 3 ⊢ ≤ ∈ V
5		eqid 2760	. . . 4 ⊢ dom ≤ = dom ≤
6		eqid 2760	. . . 4 ⊢ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥})
7		eqid 2760	. . . 4 ⊢ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦})
8	5, 6, 7	ordtval 23513	. . 3 ⊢ ( ≤ ∈ V → (ordTop‘ ≤ ) = (topGen‘(fi‘({dom ≤ } ∪ (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}))))))
9	4, 8	ax-mp 5	. 2 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘({dom ≤ } ∪ (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦})))))
10		ordtNEW.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
11	10, 1	prsdm 34593	. . . . . 6 ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵)
12	11	sneqd 4592	. . . . 5 ⊢ (𝐾 ∈ Proset → {dom ≤ } = {𝐵})
13	11	rabeqdv 3427	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
14	11, 13	mpteq12dv 5191	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
15	14	rneqd 5919	. . . . . . 7 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
16		ordtposval.e	. . . . . . 7 ⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
17	15, 16	eqtr4di 2813	. . . . . 6 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = 𝐸)
18	11	rabeqdv 3427	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
19	11, 18	mpteq12dv 5191	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
20	19	rneqd 5919	. . . . . . 7 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
21		ordtposval.f	. . . . . . 7 ⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
22	20, 21	eqtr4di 2813	. . . . . 6 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = 𝐹)
23	17, 22	uneq12d 4168	. . . . 5 ⊢ (𝐾 ∈ Proset → (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦})) = (𝐸 ∪ 𝐹))
24	12, 23	uneq12d 4168	. . . 4 ⊢ (𝐾 ∈ Proset → ({dom ≤ } ∪ (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}))) = ({𝐵} ∪ (𝐸 ∪ 𝐹)))
25	24	fveq2d 6890	. . 3 ⊢ (𝐾 ∈ Proset → (fi‘({dom ≤ } ∪ (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦})))) = (fi‘({𝐵} ∪ (𝐸 ∪ 𝐹))))
26	25	fveq2d 6890	. 2 ⊢ (𝐾 ∈ Proset → (topGen‘(fi‘({dom ≤ } ∪ (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}))))) = (topGen‘(fi‘({𝐵} ∪ (𝐸 ∪ 𝐹)))))
27	9, 26	eqtrid 2807	1 ⊢ (𝐾 ∈ Proset → (ordTop‘ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸 ∪ 𝐹)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2144  {crab 3412  Vcvv 3450   ∩ cin 4089   ∪ cun 4149  {csn 4584   class class class wbr 5102   ↦ cmpt 5185   × cxp 5648  dom cdm 5650  ran crn 5651  ‘cfv 6539  ficfi 9406  Basecbs 17393  lecple 17441  topGenctg 17614  ordTopcordt 17677   Proset cproset 18472

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fv 6547  df-ordt 17679  df-proset 18474

This theorem is used by:  ordtcnvNEW  34599  ordtrest2NEW  34602  ordtconnlem1  34603