MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtuni Structured version   Visualization version   GIF version
Theorem ordtuni 22915
Description: Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypotheses
Ref Expression
ordtval.1 𝑋 = dom 𝑅
ordtval.2 𝐴 = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
ordtval.3 𝐡 = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦})
Assertion
Ref Expression
ordtuni (𝑅 ∈ 𝑉 β†’ 𝑋 = βˆͺ ({𝑋} βˆͺ (𝐴 βˆͺ 𝐡)))
Distinct variable groups:   π‘₯,𝑦,𝑅   π‘₯,𝑋,𝑦   π‘₯,𝑉
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐡(π‘₯,𝑦)   𝑉(𝑦)
Proof of Theorem ordtuni
StepHypRef Expression
1 ordtval.1 . . . . . 6 𝑋 = dom 𝑅
2 dmexg 7897 . . . . . 6 (𝑅 ∈ 𝑉 β†’ dom 𝑅 ∈ V)
31, 2eqeltrid 2836 . . . . 5 (𝑅 ∈ 𝑉 β†’ 𝑋 ∈ V)
4 unisng 4929 . . . . 5 (𝑋 ∈ V β†’ βˆͺ {𝑋} = 𝑋)
53, 4syl 17 . . . 4 (𝑅 ∈ 𝑉 β†’ βˆͺ {𝑋} = 𝑋)
65uneq1d 4162 . . 3 (𝑅 ∈ 𝑉 β†’ (βˆͺ {𝑋} βˆͺ βˆͺ (𝐴 βˆͺ 𝐡)) = (𝑋 βˆͺ βˆͺ (𝐴 βˆͺ 𝐡)))
7 ordtval.2 . . . . . . 7 𝐴 = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯})
8 ssrab2 4077 . . . . . . . . . 10 {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} βŠ† 𝑋
93adantr 480 . . . . . . . . . . 11 ((𝑅 ∈ 𝑉 ∧ π‘₯ ∈ 𝑋) β†’ 𝑋 ∈ V)
10 elpw2g 5344 . . . . . . . . . . 11 (𝑋 ∈ V β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} βŠ† 𝑋))
119, 10syl 17 . . . . . . . . . 10 ((𝑅 ∈ 𝑉 ∧ π‘₯ ∈ 𝑋) β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} βŠ† 𝑋))
128, 11mpbiri 258 . . . . . . . . 9 ((𝑅 ∈ 𝑉 ∧ π‘₯ ∈ 𝑋) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∈ 𝒫 𝑋)
1312fmpttd 7116 . . . . . . . 8 (𝑅 ∈ 𝑉 β†’ (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}):π‘‹βŸΆπ’« 𝑋)
1413frnd 6725 . . . . . . 7 (𝑅 ∈ 𝑉 β†’ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) βŠ† 𝒫 𝑋)
157, 14eqsstrid 4030 . . . . . 6 (𝑅 ∈ 𝑉 β†’ 𝐴 βŠ† 𝒫 𝑋)
16 ordtval.3 . . . . . . 7 𝐡 = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦})
17 ssrab2 4077 . . . . . . . . . 10 {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} βŠ† 𝑋
18 elpw2g 5344 . . . . . . . . . . 11 (𝑋 ∈ V β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} βŠ† 𝑋))
199, 18syl 17 . . . . . . . . . 10 ((𝑅 ∈ 𝑉 ∧ π‘₯ ∈ 𝑋) β†’ ({𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} βŠ† 𝑋))
2017, 19mpbiri 258 . . . . . . . . 9 ((𝑅 ∈ 𝑉 ∧ π‘₯ ∈ 𝑋) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∈ 𝒫 𝑋)
2120fmpttd 7116 . . . . . . . 8 (𝑅 ∈ 𝑉 β†’ (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}):π‘‹βŸΆπ’« 𝑋)
2221frnd 6725 . . . . . . 7 (𝑅 ∈ 𝑉 β†’ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βŠ† 𝒫 𝑋)
2316, 22eqsstrid 4030 . . . . . 6 (𝑅 ∈ 𝑉 β†’ 𝐡 βŠ† 𝒫 𝑋)
2415, 23unssd 4186 . . . . 5 (𝑅 ∈ 𝑉 β†’ (𝐴 βˆͺ 𝐡) βŠ† 𝒫 𝑋)
25 sspwuni 5103 . . . . 5 ((𝐴 βˆͺ 𝐡) βŠ† 𝒫 𝑋 ↔ βˆͺ (𝐴 βˆͺ 𝐡) βŠ† 𝑋)
2624, 25sylib 217 . . . 4 (𝑅 ∈ 𝑉 β†’ βˆͺ (𝐴 βˆͺ 𝐡) βŠ† 𝑋)
27 ssequn2 4183 . . . 4 (βˆͺ (𝐴 βˆͺ 𝐡) βŠ† 𝑋 ↔ (𝑋 βˆͺ βˆͺ (𝐴 βˆͺ 𝐡)) = 𝑋)
2826, 27sylib 217 . . 3 (𝑅 ∈ 𝑉 β†’ (𝑋 βˆͺ βˆͺ (𝐴 βˆͺ 𝐡)) = 𝑋)
296, 28eqtr2d 2772 . 2 (𝑅 ∈ 𝑉 β†’ 𝑋 = (βˆͺ {𝑋} βˆͺ βˆͺ (𝐴 βˆͺ 𝐡)))
30 uniun 4934 . 2 βˆͺ ({𝑋} βˆͺ (𝐴 βˆͺ 𝐡)) = (βˆͺ {𝑋} βˆͺ βˆͺ (𝐴 βˆͺ 𝐡))
3129, 30eqtr4di 2789 1 (𝑅 ∈ 𝑉 β†’ 𝑋 = βˆͺ ({𝑋} βˆͺ (𝐴 βˆͺ 𝐡)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {crab 3431  Vcvv 3473   βˆͺ cun 3946   βŠ† wss 3948  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ran crn 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  ordtbas2  22916  ordtbas  22917  ordttopon  22918  ordtopn1  22919  ordtopn2  22920  ordtrest2  22929  ordthmeolem  23526  ordtprsuni  33198
  Copyright terms: Public domain W3C validator