Theorem ordtopn1 23256

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 2764	. . . . . . . . 9 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
3		eqid 2764	. . . . . . . . 9 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
4	1, 2, 3	ordtuni 23252	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
5	4	adantr 484	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
6		dmexg 7884	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
7	1, 6	eqeltrid 2868	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → 𝑋 ∈ V)
8	7	adantr 484	. . . . . . 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 7749	. . . . . 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 236	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
12		ssfii 9367	. . . . 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 17	. . . 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 23039	. . . . 5 ⊢ (fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ∈ TopBases
15		bastg 23028	. . . . 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 3950	. . 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 23251	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
19	18	adantr 484	. . 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 3975	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (ordTop‘𝑅))
21		ssun2 4133	. . 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 4132	. . . 4 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⊆ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
23		simpr 488	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
24		eqidd 2765	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃})
25		breq2 5106	. . . . . . . . 9 ⊢ (𝑦 = 𝑃 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑃))
26	25	notbid 320	. . . . . . . 8 ⊢ (𝑦 = 𝑃 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑃))
27	26	rabbidv 3423	. . . . . . 7 ⊢ (𝑦 = 𝑃 → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃})
28	27	rspceeqv 3606	. . . . . 6 ⊢ ((𝑃 ∈ 𝑋 ∧ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃}) → ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
29	23, 24, 28	syl2anc 593	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
30		rabexg 5295	. . . . . 6 ⊢ (𝑋 ∈ V → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ V)
31		eqid 2764	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
32	31	elrnmpt 5936	. . . . . 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 18	. . . . 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 259	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
35	22, 34	sselid 3936	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))
36	21, 35	sselid 3936	. 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 3939	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ordTop‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∃wrex 3088  {crab 3416  Vcvv 3456   ∪ cun 3904   ⊆ wss 3906  {csn 4584  ∪ cuni 4867   class class class wbr 5102   ↦ cmpt 5183  dom cdm 5649  ran crn 5650  ‘cfv 6523  ficfi 9358  topGenctg 17468  ordTopcordt 17531  TopBasesctb 23007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-om 7849  df-1o 8439  df-en 8930  df-fin 8933  df-fi 9359  df-topgen 17474  df-ordt 17533  df-bases 23008

This theorem is referenced by:  ordtopn3  23258  ordtcld1  23259  ordtrest  23264  ordtrest2lem  23265  ordthauslem  23445  ordthmeolem  23863  ordtrestNEW  34220  ordtrest2NEWlem  34221