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Theorem ordtopn2 23054
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) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} ∈ (ordTopβ€˜π‘…))
Distinct variable groups:   π‘₯,𝑃   π‘₯,𝑅   π‘₯,𝑉   π‘₯,𝑋

Proof of Theorem ordtopn2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ordttopon.3 . . . . . . . . 9 𝑋 = dom 𝑅
2 eqid 2726 . . . . . . . . 9 ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) = ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦})
3 eqid 2726 . . . . . . . . 9 ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
41, 2, 3ordtuni 23049 . . . . . . . 8 (𝑅 ∈ 𝑉 β†’ 𝑋 = βˆͺ ({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))))
54adantr 480 . . . . . . 7 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ 𝑋 = βˆͺ ({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))))
6 dmexg 7891 . . . . . . . . 9 (𝑅 ∈ 𝑉 β†’ dom 𝑅 ∈ V)
71, 6eqeltrid 2831 . . . . . . . 8 (𝑅 ∈ 𝑉 β†’ 𝑋 ∈ V)
87adantr 480 . . . . . . 7 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ 𝑋 ∈ V)
95, 8eqeltrrd 2828 . . . . . 6 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ βˆͺ ({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))) ∈ V)
10 uniexb 7748 . . . . . 6 (({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))) ∈ V ↔ βˆͺ ({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))) ∈ V)
119, 10sylibr 233 . . . . 5 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ ({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))) ∈ V)
12 ssfii 9416 . . . . 5 (({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))) ∈ V β†’ ({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))) βŠ† (fiβ€˜({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})))))
1311, 12syl 17 . . . 4 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ ({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))) βŠ† (fiβ€˜({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})))))
14 fibas 22835 . . . . 5 (fiβ€˜({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})))) ∈ TopBases
15 bastg 22824 . . . . 5 ((fiβ€˜({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})))) ∈ TopBases β†’ (fiβ€˜({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})))) βŠ† (topGenβ€˜(fiβ€˜({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))))))
1614, 15ax-mp 5 . . . 4 (fiβ€˜({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})))) βŠ† (topGenβ€˜(fiβ€˜({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})))))
1713, 16sstrdi 3989 . . 3 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ ({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))) βŠ† (topGenβ€˜(fiβ€˜({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))))))
181, 2, 3ordtval 23048 . . . 4 (𝑅 ∈ 𝑉 β†’ (ordTopβ€˜π‘…) = (topGenβ€˜(fiβ€˜({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))))))
1918adantr 480 . . 3 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ (ordTopβ€˜π‘…) = (topGenβ€˜(fiβ€˜({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))))))
2017, 19sseqtrrd 4018 . 2 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ ({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))) βŠ† (ordTopβ€˜π‘…))
21 ssun2 4168 . . 3 (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})) βŠ† ({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})))
22 ssun2 4168 . . . 4 ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) βŠ† (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
23 simpr 484 . . . . . 6 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ 𝑃 ∈ 𝑋)
24 eqidd 2727 . . . . . 6 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯})
25 breq1 5144 . . . . . . . . 9 (𝑦 = 𝑃 β†’ (𝑦𝑅π‘₯ ↔ 𝑃𝑅π‘₯))
2625notbid 318 . . . . . . . 8 (𝑦 = 𝑃 β†’ (Β¬ 𝑦𝑅π‘₯ ↔ Β¬ 𝑃𝑅π‘₯))
2726rabbidv 3434 . . . . . . 7 (𝑦 = 𝑃 β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯})
2827rspceeqv 3628 . . . . . 6 ((𝑃 ∈ 𝑋 ∧ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯}) β†’ βˆƒπ‘¦ ∈ 𝑋 {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
2923, 24, 28syl2anc 583 . . . . 5 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ βˆƒπ‘¦ ∈ 𝑋 {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
30 rabexg 5324 . . . . . 6 (𝑋 ∈ V β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} ∈ V)
31 eqid 2726 . . . . . . 7 (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
3231elrnmpt 5949 . . . . . 6 ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} ∈ V β†’ ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} ∈ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) ↔ βˆƒπ‘¦ ∈ 𝑋 {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
338, 30, 323syl 18 . . . . 5 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ ({π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} ∈ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) ↔ βˆƒπ‘¦ ∈ 𝑋 {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
3429, 33mpbird 257 . . . 4 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} ∈ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
3522, 34sselid 3975 . . 3 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} ∈ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})))
3621, 35sselid 3975 . 2 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} ∈ ({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))))
3720, 36sseldd 3978 1 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} ∈ (ordTopβ€˜π‘…))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064  {crab 3426  Vcvv 3468   βˆͺ cun 3941   βŠ† wss 3943  {csn 4623  βˆͺ cuni 4902   class class class wbr 5141   ↦ cmpt 5224  dom cdm 5669  ran crn 5670  β€˜cfv 6537  ficfi 9407  topGenctg 17392  ordTopcordt 17454  TopBasesctb 22803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7853  df-1o 8467  df-en 8942  df-fin 8945  df-fi 9408  df-topgen 17398  df-ordt 17456  df-bases 22804
This theorem is referenced by:  ordtopn3  23055  ordtcld2  23057  ordtrest  23061  ordthauslem  23242  ordthmeolem  23660  ordtrestNEW  33431
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