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Theorem ordtopn2 21733
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 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 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.
StepHypRef Expression
1 ordttopon.3 . . . . . . . . 9 𝑋 = dom 𝑅
2 eqid 2821 . . . . . . . . 9 ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦})
3 eqid 2821 . . . . . . . . 9 ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
41, 2, 3ordtuni 21728 . . . . . . . 8 (𝑅𝑉𝑋 = ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
54adantr 481 . . . . . . 7 ((𝑅𝑉𝑃𝑋) → 𝑋 = ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
6 dmexg 7601 . . . . . . . . 9 (𝑅𝑉 → dom 𝑅 ∈ V)
71, 6eqeltrid 2917 . . . . . . . 8 (𝑅𝑉𝑋 ∈ V)
87adantr 481 . . . . . . 7 ((𝑅𝑉𝑃𝑋) → 𝑋 ∈ V)
95, 8eqeltrrd 2914 . . . . . 6 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
10 uniexb 7474 . . . . . 6 (({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V ↔ ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
119, 10sylibr 235 . . . . 5 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
12 ssfii 8872 . . . . 5 (({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))))
1311, 12syl 17 . . . 4 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))))
14 fibas 21515 . . . . 5 (fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ∈ TopBases
15 bastg 21504 . . . . 5 ((fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ∈ TopBases → (fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ⊆ (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
1614, 15ax-mp 5 . . . 4 (fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ⊆ (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))))
1713, 16sstrdi 3978 . . 3 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
181, 2, 3ordtval 21727 . . . 4 (𝑅𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
1918adantr 481 . . 3 ((𝑅𝑉𝑃𝑋) → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
2017, 19sseqtrrd 4007 . 2 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (ordTop‘𝑅))
21 ssun2 4148 . . 3 (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})) ⊆ ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))
22 ssun2 4148 . . . 4 ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))
23 simpr 485 . . . . . 6 ((𝑅𝑉𝑃𝑋) → 𝑃𝑋)
24 eqidd 2822 . . . . . 6 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥})
25 breq1 5061 . . . . . . . . 9 (𝑦 = 𝑃 → (𝑦𝑅𝑥𝑃𝑅𝑥))
2625notbid 319 . . . . . . . 8 (𝑦 = 𝑃 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑃𝑅𝑥))
2726rabbidv 3481 . . . . . . 7 (𝑦 = 𝑃 → {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥})
2827rspceeqv 3637 . . . . . 6 ((𝑃𝑋 ∧ {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥}) → ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
2923, 24, 28syl2anc 584 . . . . 5 ((𝑅𝑉𝑃𝑋) → ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
30 rabexg 5226 . . . . . 6 (𝑋 ∈ V → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ V)
31 eqid 2821 . . . . . . 7 (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
3231elrnmpt 5822 . . . . . 6 ({𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ V → ({𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))
338, 30, 323syl 18 . . . . 5 ((𝑅𝑉𝑃𝑋) → ({𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))
3429, 33mpbird 258 . . . 4 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))
3522, 34sseldi 3964 . . 3 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))
3621, 35sseldi 3964 . 2 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
3720, 36sseldd 3967 1 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3139  {crab 3142  Vcvv 3495  cun 3933  wss 3935  {csn 4559   cuni 4832   class class class wbr 5058  cmpt 5138  dom cdm 5549  ran crn 5550  cfv 6349  ficfi 8863  topGenctg 16701  ordTopcordt 16762  TopBasesctb 21483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-fin 8502  df-fi 8864  df-topgen 16707  df-ordt 16764  df-bases 21484
This theorem is referenced by:  ordtopn3  21734  ordtcld2  21736  ordtrest  21740  ordthauslem  21921  ordthmeolem  22339  ordtrestNEW  31064
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