Theorem ordtopn2 23513

Description: A downward ray (-∞, 𝑃) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ordttopon.3	⊢ 𝑋 = dom 𝑅

Assertion

Ref	Expression
ordtopn2	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅))

Distinct variable groups:   𝑥,𝑃   𝑥,𝑅   𝑥,𝑉   𝑥,𝑋

Proof of Theorem ordtopn2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordttopon.3	. . . . . . . . 9 ⊢ 𝑋 = dom 𝑅
2		eqid 2761	. . . . . . . . 9 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
3		eqid 2761	. . . . . . . . 9 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
4	1, 2, 3	ordtuni 23508	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
5	4	adantr 486	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
6		dmexg 7913	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
7	1, 6	eqeltrid 2865	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → 𝑋 ∈ V)
8	7	adantr 486	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → 𝑋 ∈ V)
9	5, 8	eqeltrrd 2862	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ∪ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
10		uniexb 7778	. . . . . 6 ⊢ (({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V ↔ ∪ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
11	9, 10	sylibr 237	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
12		ssfii 9411	. . . . 5 ⊢ (({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V → ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))))
13	11, 12	syl 18	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))))
14		fibas 23295	. . . . 5 ⊢ (fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ∈ TopBases
15		bastg 23284	. . . . 5 ⊢ ((fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ∈ TopBases → (fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ⊆ (topGen‘(fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
16	14, 15	ax-mp 5	. . . 4 ⊢ (fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ⊆ (topGen‘(fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))))
17	13, 16	sstrdi 3943	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (topGen‘(fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
18	1, 2, 3	ordtval 23507	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
19	18	adantr 486	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
20	17, 19	sseqtrrd 3968	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (ordTop‘𝑅))
21		ssun2 4125	. . 3 ⊢ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})) ⊆ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))
22		ssun2 4125	. . . 4 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
23		simpr 490	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
24		eqidd 2762	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥})
25		breq1 5106	. . . . . . . . 9 ⊢ (𝑦 = 𝑃 → (𝑦𝑅𝑥 ↔ 𝑃𝑅𝑥))
26	25	notbid 321	. . . . . . . 8 ⊢ (𝑦 = 𝑃 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑃𝑅𝑥))
27	26	rabbidv 3420	. . . . . . 7 ⊢ (𝑦 = 𝑃 → {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥})
28	27	rspceeqv 3599	. . . . . 6 ⊢ ((𝑃 ∈ 𝑋 ∧ {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥}) → ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
29	23, 24, 28	syl2anc 596	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
30		rabexg 5299	. . . . . 6 ⊢ (𝑋 ∈ V → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ V)
31		eqid 2761	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
32	31	elrnmpt 5940	. . . . . 6 ⊢ ({𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ V → ({𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
33	8, 30, 32	3syl 19	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
34	29, 33	mpbird 260	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
35	22, 34	sselid 3929	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))
36	21, 35	sselid 3929	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
37	20, 36	sseldd 3932	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  {crab 3413  Vcvv 3451   ∪ cun 3897   ⊆ wss 3899  {csn 4584  ∪ cuni 4867   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5651  ran crn 5652  ‘cfv 6538  ficfi 9402  topGenctg 17608  ordTopcordt 17671  TopBasesctb 23263

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8476  df-en 8974  df-fin 8977  df-fi 9403  df-topgen 17614  df-ordt 17673  df-bases 23264

This theorem is used by:  ordtopn3  23514  ordtcld2  23516  ordtrest  23520  ordthauslem  23701  ordthmeolem  24120  ordtrestNEW  34553