Theorem ordtopn1 22345

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 2738	. . . . . . . . 9 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
3		eqid 2738	. . . . . . . . 9 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
4	1, 2, 3	ordtuni 22341	. . . . . . . 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 7750	. . . . . . . . 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 7614	. . . . . 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 233	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
12		ssfii 9178	. . . . 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 22127	. . . . 5 ⊢ (fi‘({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ∈ TopBases
15		bastg 22116	. . . . 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 3933	. . 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 22340	. . . 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 3962	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ({𝑋} ∪ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (ordTop‘𝑅))
21		ssun2 4107	. . 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 4106	. . . 4 ⊢ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⊆ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
23		simpr 485	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
24		eqidd 2739	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃})
25		breq2 5078	. . . . . . . . 9 ⊢ (𝑦 = 𝑃 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑃))
26	25	notbid 318	. . . . . . . 8 ⊢ (𝑦 = 𝑃 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑃))
27	26	rabbidv 3414	. . . . . . 7 ⊢ (𝑦 = 𝑃 → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃})
28	27	rspceeqv 3575	. . . . . 6 ⊢ ((𝑃 ∈ 𝑋 ∧ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃}) → ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
29	23, 24, 28	syl2anc 584	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
30		rabexg 5255	. . . . . 6 ⊢ (𝑋 ∈ V → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ V)
31		eqid 2738	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
32	31	elrnmpt 5865	. . . . . 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 256	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
35	22, 34	sselid 3919	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦 ∈ 𝑋 ↦ {𝑥 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))
36	21, 35	sselid 3919	. 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 3922	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ordTop‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  {crab 3068  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  {csn 4561  ∪ cuni 4839   class class class wbr 5074   ↦ cmpt 5157  dom cdm 5589  ran crn 5590  ‘cfv 6433  ficfi 9169  topGenctg 17148  ordTopcordt 17210  TopBasesctb 22095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-en 8734  df-fin 8737  df-fi 9170  df-topgen 17154  df-ordt 17212  df-bases 22096

This theorem is referenced by:  ordtopn3  22347  ordtcld1  22348  ordtrest  22353  ordtrest2lem  22354  ordthauslem  22534  ordthmeolem  22952  ordtrestNEW  31871  ordtrest2NEWlem  31872