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Theorem ordtopn1 21220
 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 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 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.
StepHypRef Expression
1 ordttopon.3 . . . . . . . . 9 𝑋 = dom 𝑅
2 eqid 2760 . . . . . . . . 9 ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦})
3 eqid 2760 . . . . . . . . 9 ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
41, 2, 3ordtuni 21216 . . . . . . . 8 (𝑅𝑉𝑋 = ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
54adantr 472 . . . . . . 7 ((𝑅𝑉𝑃𝑋) → 𝑋 = ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
6 dmexg 7263 . . . . . . . . 9 (𝑅𝑉 → dom 𝑅 ∈ V)
71, 6syl5eqel 2843 . . . . . . . 8 (𝑅𝑉𝑋 ∈ V)
87adantr 472 . . . . . . 7 ((𝑅𝑉𝑃𝑋) → 𝑋 ∈ V)
95, 8eqeltrrd 2840 . . . . . 6 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
10 uniexb 7139 . . . . . 6 (({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V ↔ ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
119, 10sylibr 224 . . . . 5 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
12 ssfii 8492 . . . . 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 21003 . . . . 5 (fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ∈ TopBases
15 bastg 20992 . . . . 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, 16syl6ss 3756 . . 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 21215 . . . 4 (𝑅𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
1918adantr 472 . . 3 ((𝑅𝑉𝑃𝑋) → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
2017, 19sseqtr4d 3783 . 2 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (ordTop‘𝑅))
21 ssun2 3920 . . 3 (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})) ⊆ ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))
22 ssun1 3919 . . . 4 ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⊆ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))
23 simpr 479 . . . . . 6 ((𝑅𝑉𝑃𝑋) → 𝑃𝑋)
24 eqidd 2761 . . . . . 6 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃})
25 breq2 4808 . . . . . . . . . 10 (𝑦 = 𝑃 → (𝑥𝑅𝑦𝑥𝑅𝑃))
2625notbid 307 . . . . . . . . 9 (𝑦 = 𝑃 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑃))
2726rabbidv 3329 . . . . . . . 8 (𝑦 = 𝑃 → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃})
2827eqeq2d 2770 . . . . . . 7 (𝑦 = 𝑃 → ({𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦} ↔ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃}))
2928rspcev 3449 . . . . . 6 ((𝑃𝑋 ∧ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃}) → ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦})
3023, 24, 29syl2anc 696 . . . . 5 ((𝑅𝑉𝑃𝑋) → ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦})
31 rabexg 4963 . . . . . 6 (𝑋 ∈ V → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ V)
32 eqid 2760 . . . . . . 7 (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦})
3332elrnmpt 5527 . . . . . 6 ({𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ V → ({𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}))
348, 31, 333syl 18 . . . . 5 ((𝑅𝑉𝑃𝑋) → ({𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} = {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}))
3530, 34mpbird 247 . . . 4 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}))
3622, 35sseldi 3742 . . 3 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))
3721, 36sseldi 3742 . 2 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
3820, 37sseldd 3745 1 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑃} ∈ (ordTop‘𝑅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∃wrex 3051  {crab 3054  Vcvv 3340   ∪ cun 3713   ⊆ wss 3715  {csn 4321  ∪ cuni 4588   class class class wbr 4804   ↦ cmpt 4881  dom cdm 5266  ran crn 5267  ‘cfv 6049  ficfi 8483  topGenctg 16320  ordTopcordt 16381  TopBasesctb 20971 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-fin 8127  df-fi 8484  df-topgen 16326  df-ordt 16383  df-bases 20972 This theorem is referenced by:  ordtopn3  21222  ordtcld1  21223  ordtrest  21228  ordtrest2lem  21229  ordthauslem  21409  ordthmeolem  21826  ordtrestNEW  30297  ordtrest2NEWlem  30298
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