Theorem ordtopn2 23178

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 2739	. . . . . . . . 9 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
3		eqid 2739	. . . . . . . . 9 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
4	1, 2, 3	ordtuni 23173	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
5	4	adantr 481	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → 𝑋 = ∪ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
6		dmexg 7841	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
7	1, 6	eqeltrid 2843	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → 𝑋 ∈ V)
8	7	adantr 481	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → 𝑋 ∈ V)
9	5, 8	eqeltrrd 2840	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ∪ ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
10		uniexb 7707	. . . . . 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 235	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
12		ssfii 9322	. . . . 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 22960	. . . . 5 ⊢ (fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ∈ TopBases
15		bastg 22949	. . . . 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 3927	. . 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 23172	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
19	18	adantr 481	. . 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 3952	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (ordTop‘𝑅))
21		ssun2 4108	. . 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 4108	. . . 4 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
23		simpr 485	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
24		eqidd 2740	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥})
25		breq1 5075	. . . . . . . . 9 ⊢ (𝑦 = 𝑃 → (𝑦𝑅𝑥 ↔ 𝑃𝑅𝑥))
26	25	notbid 319	. . . . . . . 8 ⊢ (𝑦 = 𝑃 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑃𝑅𝑥))
27	26	rabbidv 3398	. . . . . . 7 ⊢ (𝑦 = 𝑃 → {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥})
28	27	rspceeqv 3583	. . . . . 6 ⊢ ((𝑃 ∈ 𝑋 ∧ {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥}) → ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
29	23, 24, 28	syl2anc 590	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
30		rabexg 5265	. . . . . 6 ⊢ (𝑋 ∈ V → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ V)
31		eqid 2739	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
32	31	elrnmpt 5900	. . . . . 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 18	. . . . 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 258	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
35	22, 34	sselid 3913	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))
36	21, 35	sselid 3913	. 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 3916	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  {crab 3391  Vcvv 3431   ∪ cun 3881   ⊆ wss 3883  {csn 4555  ∪ cuni 4838   class class class wbr 5072   ↦ cmpt 5153  dom cdm 5618  ran crn 5619  ‘cfv 6485  ficfi 9313  topGenctg 17391  ordTopcordt 17454  TopBasesctb 22928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-om 7807  df-1o 8395  df-en 8884  df-fin 8887  df-fi 9314  df-topgen 17397  df-ordt 17456  df-bases 22929

This theorem is referenced by:  ordtopn3  23179  ordtcld2  23181  ordtrest  23185  ordthauslem  23366  ordthmeolem  23784  ordtrestNEW  34105