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Theorem ordtuni 21726
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 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
ordtval.3 𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
Assertion
Ref Expression
ordtuni (𝑅𝑉𝑋 = ({𝑋} ∪ (𝐴𝐵)))
Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑋,𝑦   𝑥,𝑉
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem ordtuni
StepHypRef Expression
1 ordtval.1 . . . . . 6 𝑋 = dom 𝑅
2 dmexg 7602 . . . . . 6 (𝑅𝑉 → dom 𝑅 ∈ V)
31, 2eqeltrid 2914 . . . . 5 (𝑅𝑉𝑋 ∈ V)
4 unisng 4845 . . . . 5 (𝑋 ∈ V → {𝑋} = 𝑋)
53, 4syl 17 . . . 4 (𝑅𝑉 {𝑋} = 𝑋)
65uneq1d 4135 . . 3 (𝑅𝑉 → ( {𝑋} ∪ (𝐴𝐵)) = (𝑋 (𝐴𝐵)))
7 ordtval.2 . . . . . . 7 𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
8 ssrab2 4053 . . . . . . . . . 10 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
93adantr 481 . . . . . . . . . . 11 ((𝑅𝑉𝑥𝑋) → 𝑋 ∈ V)
10 elpw2g 5238 . . . . . . . . . . 11 (𝑋 ∈ V → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
119, 10syl 17 . . . . . . . . . 10 ((𝑅𝑉𝑥𝑋) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
128, 11mpbiri 259 . . . . . . . . 9 ((𝑅𝑉𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋)
1312fmpttd 6871 . . . . . . . 8 (𝑅𝑉 → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
1413frnd 6514 . . . . . . 7 (𝑅𝑉 → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
157, 14eqsstrid 4012 . . . . . 6 (𝑅𝑉𝐴 ⊆ 𝒫 𝑋)
16 ordtval.3 . . . . . . 7 𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
17 ssrab2 4053 . . . . . . . . . 10 {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋
18 elpw2g 5238 . . . . . . . . . . 11 (𝑋 ∈ V → ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋))
199, 18syl 17 . . . . . . . . . 10 ((𝑅𝑉𝑥𝑋) → ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋))
2017, 19mpbiri 259 . . . . . . . . 9 ((𝑅𝑉𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ 𝒫 𝑋)
2120fmpttd 6871 . . . . . . . 8 (𝑅𝑉 → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}):𝑋⟶𝒫 𝑋)
2221frnd 6514 . . . . . . 7 (𝑅𝑉 → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⊆ 𝒫 𝑋)
2316, 22eqsstrid 4012 . . . . . 6 (𝑅𝑉𝐵 ⊆ 𝒫 𝑋)
2415, 23unssd 4159 . . . . 5 (𝑅𝑉 → (𝐴𝐵) ⊆ 𝒫 𝑋)
25 sspwuni 5013 . . . . 5 ((𝐴𝐵) ⊆ 𝒫 𝑋 (𝐴𝐵) ⊆ 𝑋)
2624, 25sylib 219 . . . 4 (𝑅𝑉 (𝐴𝐵) ⊆ 𝑋)
27 ssequn2 4156 . . . 4 ( (𝐴𝐵) ⊆ 𝑋 ↔ (𝑋 (𝐴𝐵)) = 𝑋)
2826, 27sylib 219 . . 3 (𝑅𝑉 → (𝑋 (𝐴𝐵)) = 𝑋)
296, 28eqtr2d 2854 . 2 (𝑅𝑉𝑋 = ( {𝑋} ∪ (𝐴𝐵)))
30 uniun 4849 . 2 ({𝑋} ∪ (𝐴𝐵)) = ( {𝑋} ∪ (𝐴𝐵))
3129, 30syl6eqr 2871 1 (𝑅𝑉𝑋 = ({𝑋} ∪ (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  cun 3931  wss 3933  𝒫 cpw 4535  {csn 4557   cuni 4830   class class class wbr 5057  cmpt 5137  dom cdm 5548  ran crn 5549
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 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  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-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  ordtbas2  21727  ordtbas  21728  ordttopon  21729  ordtopn1  21730  ordtopn2  21731  ordtrest2  21740  ordthmeolem  22337  ordtprsuni  31061
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