Theorem ordtprsval 33885

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 6835	. . . . 5 ⊢ (le‘𝐾) ∈ V
3	2	inex1 5256	. . . 4 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
4	1, 3	eqeltri 2824	. . 3 ⊢ ≤ ∈ V
5		eqid 2729	. . . 4 ⊢ dom ≤ = dom ≤
6		eqid 2729	. . . 4 ⊢ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥})
7		eqid 2729	. . . 4 ⊢ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦})
8	5, 6, 7	ordtval 23074	. . 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 33881	. . . . . 6 ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵)
12	11	sneqd 4589	. . . . 5 ⊢ (𝐾 ∈ Proset → {dom ≤ } = {𝐵})
13	11	rabeqdv 3410	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
14	11, 13	mpteq12dv 5179	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
15	14	rneqd 5880	. . . . . . 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 2782	. . . . . 6 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) = 𝐸)
18	11	rabeqdv 3410	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
19	11, 18	mpteq12dv 5179	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
20	19	rneqd 5880	. . . . . . 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 2782	. . . . . 6 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}) = 𝐹)
23	17, 22	uneq12d 4120	. . . . 5 ⊢ (𝐾 ∈ Proset → (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦})) = (𝐸 ∪ 𝐹))
24	12, 23	uneq12d 4120	. . . 4 ⊢ (𝐾 ∈ Proset → ({dom ≤ } ∪ (ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ dom ≤ ↦ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦}))) = ({𝐵} ∪ (𝐸 ∪ 𝐹)))
25	24	fveq2d 6826	. . 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 6826	. 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 2776	1 ⊢ (𝐾 ∈ Proset → (ordTop‘ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸 ∪ 𝐹)))))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3394  Vcvv 3436   ∪ cun 3901   ∩ cin 3902  {csn 4577   class class class wbr 5092   ↦ cmpt 5173   × cxp 5617  dom cdm 5619  ran crn 5620  ‘cfv 6482  ficfi 9300  Basecbs 17120  lecple 17168  topGenctg 17341  ordTopcordt 17403   Proset cproset 18198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fv 6490  df-ordt 17405  df-proset 18200

This theorem is referenced by:  ordtcnvNEW  33887  ordtrest2NEW  33890  ordtconnlem1  33891