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Theorem ordtopn2 23127
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 2728 . . . . . . . . 9 ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) = ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦})
3 eqid 2728 . . . . . . . . 9 ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
41, 2, 3ordtuni 23122 . . . . . . . 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 479 . . . . . . 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 7917 . . . . . . . . 9 (𝑅 ∈ 𝑉 β†’ dom 𝑅 ∈ V)
71, 6eqeltrid 2833 . . . . . . . 8 (𝑅 ∈ 𝑉 β†’ 𝑋 ∈ V)
87adantr 479 . . . . . . 7 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ 𝑋 ∈ V)
95, 8eqeltrrd 2830 . . . . . 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 7774 . . . . . 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 9452 . . . . 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 22908 . . . . 5 (fiβ€˜({𝑋} βˆͺ (ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})))) ∈ TopBases
15 bastg 22897 . . . . 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 3994 . . 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 23121 . . . 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 479 . . 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 4023 . 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 4175 . . 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 4175 . . . 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 483 . . . . . 6 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ 𝑃 ∈ 𝑋)
24 eqidd 2729 . . . . . 6 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯})
25 breq1 5155 . . . . . . . . 9 (𝑦 = 𝑃 β†’ (𝑦𝑅π‘₯ ↔ 𝑃𝑅π‘₯))
2625notbid 317 . . . . . . . 8 (𝑦 = 𝑃 β†’ (Β¬ 𝑦𝑅π‘₯ ↔ Β¬ 𝑃𝑅π‘₯))
2726rabbidv 3438 . . . . . . 7 (𝑦 = 𝑃 β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯})
2827rspceeqv 3633 . . . . . 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 582 . . . . 5 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ βˆƒπ‘¦ ∈ 𝑋 {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} = {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
30 rabexg 5337 . . . . . 6 (𝑋 ∈ V β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} ∈ V)
31 eqid 2728 . . . . . . 7 (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) = (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
3231elrnmpt 5962 . . . . . 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 256 . . . 4 ((𝑅 ∈ 𝑉 ∧ 𝑃 ∈ 𝑋) β†’ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑃𝑅π‘₯} ∈ ran (𝑦 ∈ 𝑋 ↦ {π‘₯ ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}))
3522, 34sselid 3980 . . 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 3980 . 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 3983 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 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067  {crab 3430  Vcvv 3473   βˆͺ cun 3947   βŠ† wss 3949  {csn 4632  βˆͺ cuni 4912   class class class wbr 5152   ↦ cmpt 5235  dom cdm 5682  ran crn 5683  β€˜cfv 6553  ficfi 9443  topGenctg 17428  ordTopcordt 17490  TopBasesctb 22876
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7879  df-1o 8495  df-en 8973  df-fin 8976  df-fi 9444  df-topgen 17434  df-ordt 17492  df-bases 22877
This theorem is referenced by:  ordtopn3  23128  ordtcld2  23130  ordtrest  23134  ordthauslem  23315  ordthmeolem  23733  ordtrestNEW  33563
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