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Theorem ordtuni 22914
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 7896 . . . . . 6 (𝑅 ∈ 𝑉 β†’ dom 𝑅 ∈ V)
31, 2eqeltrid 2835 . . . . 5 (𝑅 ∈ 𝑉 β†’ 𝑋 ∈ V)
4 unisng 4928 . . . . 5 (𝑋 ∈ V β†’ βˆͺ {𝑋} = 𝑋)
53, 4syl 17 . . . 4 (𝑅 ∈ 𝑉 β†’ βˆͺ {𝑋} = 𝑋)
65uneq1d 4161 . . 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 4076 . . . . . . . . . 10 {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} βŠ† 𝑋
93adantr 479 . . . . . . . . . . 11 ((𝑅 ∈ 𝑉 ∧ π‘₯ ∈ 𝑋) β†’ 𝑋 ∈ V)
10 elpw2g 5343 . . . . . . . . . . 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 257 . . . . . . . . 9 ((𝑅 ∈ 𝑉 ∧ π‘₯ ∈ 𝑋) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯} ∈ 𝒫 𝑋)
1312fmpttd 7115 . . . . . . . 8 (𝑅 ∈ 𝑉 β†’ (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}):π‘‹βŸΆπ’« 𝑋)
1413frnd 6724 . . . . . . 7 (𝑅 ∈ 𝑉 β†’ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ 𝑦𝑅π‘₯}) βŠ† 𝒫 𝑋)
157, 14eqsstrid 4029 . . . . . 6 (𝑅 ∈ 𝑉 β†’ 𝐴 βŠ† 𝒫 𝑋)
16 ordtval.3 . . . . . . 7 𝐡 = ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦})
17 ssrab2 4076 . . . . . . . . . 10 {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} βŠ† 𝑋
18 elpw2g 5343 . . . . . . . . . . 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 257 . . . . . . . . 9 ((𝑅 ∈ 𝑉 ∧ π‘₯ ∈ 𝑋) β†’ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦} ∈ 𝒫 𝑋)
2120fmpttd 7115 . . . . . . . 8 (𝑅 ∈ 𝑉 β†’ (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}):π‘‹βŸΆπ’« 𝑋)
2221frnd 6724 . . . . . . 7 (𝑅 ∈ 𝑉 β†’ ran (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ Β¬ π‘₯𝑅𝑦}) βŠ† 𝒫 𝑋)
2316, 22eqsstrid 4029 . . . . . 6 (𝑅 ∈ 𝑉 β†’ 𝐡 βŠ† 𝒫 𝑋)
2415, 23unssd 4185 . . . . 5 (𝑅 ∈ 𝑉 β†’ (𝐴 βˆͺ 𝐡) βŠ† 𝒫 𝑋)
25 sspwuni 5102 . . . . 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 4182 . . . 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 2771 . 2 (𝑅 ∈ 𝑉 β†’ 𝑋 = (βˆͺ {𝑋} βˆͺ βˆͺ (𝐴 βˆͺ 𝐡)))
30 uniun 4933 . 2 βˆͺ ({𝑋} βˆͺ (𝐴 βˆͺ 𝐡)) = (βˆͺ {𝑋} βˆͺ βˆͺ (𝐴 βˆͺ 𝐡))
3129, 30eqtr4di 2788 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 394   = wceq 1539   ∈ wcel 2104  {crab 3430  Vcvv 3472   βˆͺ cun 3945   βŠ† wss 3947  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5675  ran crn 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-fun 6544  df-fn 6545  df-f 6546
This theorem is referenced by:  ordtbas2  22915  ordtbas  22916  ordttopon  22917  ordtopn1  22918  ordtopn2  22919  ordtrest2  22928  ordthmeolem  23525  ordtprsuni  33197
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