MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtopn2 Structured version   Visualization version   GIF version

Theorem ordtopn2 21795
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 2819 . . . . . . . . 9 ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦})
3 eqid 2819 . . . . . . . . 9 ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
41, 2, 3ordtuni 21790 . . . . . . . 8 (𝑅𝑉𝑋 = ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
54adantr 483 . . . . . . 7 ((𝑅𝑉𝑃𝑋) → 𝑋 = ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
6 dmexg 7605 . . . . . . . . 9 (𝑅𝑉 → dom 𝑅 ∈ V)
71, 6eqeltrid 2915 . . . . . . . 8 (𝑅𝑉𝑋 ∈ V)
87adantr 483 . . . . . . 7 ((𝑅𝑉𝑃𝑋) → 𝑋 ∈ V)
95, 8eqeltrrd 2912 . . . . . 6 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
10 uniexb 7478 . . . . . 6 (({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V ↔ ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
119, 10sylibr 236 . . . . 5 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
12 ssfii 8875 . . . . 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 21577 . . . . 5 (fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ∈ TopBases
15 bastg 21566 . . . . 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 3977 . . 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 21789 . . . 4 (𝑅𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
1918adantr 483 . . 3 ((𝑅𝑉𝑃𝑋) → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
2017, 19sseqtrrd 4006 . 2 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (ordTop‘𝑅))
21 ssun2 4147 . . 3 (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})) ⊆ ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))
22 ssun2 4147 . . . 4 ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))
23 simpr 487 . . . . . 6 ((𝑅𝑉𝑃𝑋) → 𝑃𝑋)
24 eqidd 2820 . . . . . 6 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥})
25 breq1 5060 . . . . . . . . 9 (𝑦 = 𝑃 → (𝑦𝑅𝑥𝑃𝑅𝑥))
2625notbid 320 . . . . . . . 8 (𝑦 = 𝑃 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑃𝑅𝑥))
2726rabbidv 3479 . . . . . . 7 (𝑦 = 𝑃 → {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥})
2827rspceeqv 3636 . . . . . 6 ((𝑃𝑋 ∧ {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥}) → ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
2923, 24, 28syl2anc 586 . . . . 5 ((𝑅𝑉𝑃𝑋) → ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
30 rabexg 5225 . . . . . 6 (𝑋 ∈ V → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ V)
31 eqid 2819 . . . . . . 7 (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
3231elrnmpt 5821 . . . . . 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 259 . . . 4 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))
3522, 34sseldi 3963 . . 3 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))
3621, 35sseldi 3963 . 2 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
3720, 36sseldd 3966 1 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1531  wcel 2108  wrex 3137  {crab 3140  Vcvv 3493  cun 3932  wss 3934  {csn 4559   cuni 4830   class class class wbr 5057  cmpt 5137  dom cdm 5548  ran crn 5549  cfv 6348  ficfi 8866  topGenctg 16703  ordTopcordt 16764  TopBasesctb 21545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-en 8502  df-fin 8505  df-fi 8867  df-topgen 16709  df-ordt 16766  df-bases 21546
This theorem is referenced by:  ordtopn3  21796  ordtcld2  21798  ordtrest  21802  ordthauslem  21983  ordthmeolem  22401  ordtrestNEW  31157
  Copyright terms: Public domain W3C validator