Theorem ordtopn1 23539

Description: An upward ray (𝑃, +∞) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ordttopon.3	⊢ 𝑋 = dom 𝑅

Assertion

Ref	Expression
ordtopn1	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ordTop‘𝑅))

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

Proof of Theorem ordtopn1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordttopon.3	. . . . . . . . 9 ⊢ 𝑋 = dom 𝑅
2		eqid 2763	. . . . . . . . 9 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
3		eqid 2763	. . . . . . . . 9 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
4	1, 2, 3	ordtuni 23535	. . . . . . . 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 7934	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
7	1, 6	eqeltrid 2868	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → 𝑋 ∈ V)
8	7	adantr 486	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → 𝑋 ∈ V)
9	5, 8	eqeltrrd 2865	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ∪ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
10		uniexb 7799	. . . . . 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 9435	. . . . 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 23322	. . . . 5 ⊢ (fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ∈ TopBases
15		bastg 23311	. . . . 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 3925	. . 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 23534	. . . 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 3950	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (ordTop‘𝑅))
21		ssun2 4180	. . 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		ssun1 4179	. . . 4 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⊆ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
23		simpr 490	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
24		eqidd 2764	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃})
25		breq2 5110	. . . . . . . . 9 ⊢ (𝑦 = 𝑃 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑃))
26	25	notbid 321	. . . . . . . 8 ⊢ (𝑦 = 𝑃 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑃))
27	26	rabbidv 3423	. . . . . . 7 ⊢ (𝑦 = 𝑃 → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃})
28	27	rspceeqv 3602	. . . . . 6 ⊢ ((𝑃 ∈ 𝑋 ∧ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃}) → ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
29	23, 24, 28	syl2anc 596	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
30		rabexg 5303	. . . . . 6 ⊢ (𝑋 ∈ V → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ V)
31		eqid 2763	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
32	31	elrnmpt 5946	. . . . . 6 ⊢ ({𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ V → ({𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
33	8, 30, 32	3syl 19	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
34	29, 33	mpbird 260	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
35	22, 34	sselid 3911	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))
36	21, 35	sselid 3911	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
37	20, 36	sseldd 3914	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ordTop‘𝑅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  {csn 4588  ∪ cuni 4870   class class class wbr 5106   ↦ cmpt 5189  dom cdm 5655  ran crn 5656  ‘cfv 6555  ficfi 9426  topGenctg 17634  ordTopcordt 17697  TopBasesctb 23290

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-om 7899  df-1o 8501  df-en 8999  df-fin 9002  df-fi 9427  df-topgen 17640  df-ordt 17699  df-bases 23291

This theorem is used by:  ordtopn3  23541  ordtcld1  23542  ordtrest  23547  ordtrest2lem  23548  ordthauslem  23728  ordthmeolem  24147  ordtrestNEW  34620  ordtrest2NEWlem  34621