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Theorem ordtuni 21042
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 7139 . . . . . 6 (𝑅𝑉 → dom 𝑅 ∈ V)
31, 2syl5eqel 2734 . . . . 5 (𝑅𝑉𝑋 ∈ V)
4 unisng 4484 . . . . 5 (𝑋 ∈ V → {𝑋} = 𝑋)
53, 4syl 17 . . . 4 (𝑅𝑉 {𝑋} = 𝑋)
65uneq1d 3799 . . 3 (𝑅𝑉 → ( {𝑋} ∪ (𝐴𝐵)) = (𝑋 (𝐴𝐵)))
7 ordtval.2 . . . . . . 7 𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
8 ssrab2 3720 . . . . . . . . . 10 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
93adantr 480 . . . . . . . . . . 11 ((𝑅𝑉𝑥𝑋) → 𝑋 ∈ V)
10 elpw2g 4857 . . . . . . . . . . 11 (𝑋 ∈ V → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
119, 10syl 17 . . . . . . . . . 10 ((𝑅𝑉𝑥𝑋) → ({𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
128, 11mpbiri 248 . . . . . . . . 9 ((𝑅𝑉𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋)
13 eqid 2651 . . . . . . . . 9 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
1412, 13fmptd 6425 . . . . . . . 8 (𝑅𝑉 → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
15 frn 6091 . . . . . . . 8 ((𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋 → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
1614, 15syl 17 . . . . . . 7 (𝑅𝑉 → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
177, 16syl5eqss 3682 . . . . . 6 (𝑅𝑉𝐴 ⊆ 𝒫 𝑋)
18 ordtval.3 . . . . . . 7 𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
19 ssrab2 3720 . . . . . . . . . 10 {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋
20 elpw2g 4857 . . . . . . . . . . 11 (𝑋 ∈ V → ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋))
219, 20syl 17 . . . . . . . . . 10 ((𝑅𝑉𝑥𝑋) → ({𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋))
2219, 21mpbiri 248 . . . . . . . . 9 ((𝑅𝑉𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ 𝒫 𝑋)
23 eqid 2651 . . . . . . . . 9 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
2422, 23fmptd 6425 . . . . . . . 8 (𝑅𝑉 → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}):𝑋⟶𝒫 𝑋)
25 frn 6091 . . . . . . . 8 ((𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}):𝑋⟶𝒫 𝑋 → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⊆ 𝒫 𝑋)
2624, 25syl 17 . . . . . . 7 (𝑅𝑉 → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⊆ 𝒫 𝑋)
2718, 26syl5eqss 3682 . . . . . 6 (𝑅𝑉𝐵 ⊆ 𝒫 𝑋)
2817, 27unssd 3822 . . . . 5 (𝑅𝑉 → (𝐴𝐵) ⊆ 𝒫 𝑋)
29 sspwuni 4643 . . . . 5 ((𝐴𝐵) ⊆ 𝒫 𝑋 (𝐴𝐵) ⊆ 𝑋)
3028, 29sylib 208 . . . 4 (𝑅𝑉 (𝐴𝐵) ⊆ 𝑋)
31 ssequn2 3819 . . . 4 ( (𝐴𝐵) ⊆ 𝑋 ↔ (𝑋 (𝐴𝐵)) = 𝑋)
3230, 31sylib 208 . . 3 (𝑅𝑉 → (𝑋 (𝐴𝐵)) = 𝑋)
336, 32eqtr2d 2686 . 2 (𝑅𝑉𝑋 = ( {𝑋} ∪ (𝐴𝐵)))
34 uniun 4488 . 2 ({𝑋} ∪ (𝐴𝐵)) = ( {𝑋} ∪ (𝐴𝐵))
3533, 34syl6eqr 2703 1 (𝑅𝑉𝑋 = ({𝑋} ∪ (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  cun 3605  wss 3607  𝒫 cpw 4191  {csn 4210   cuni 4468   class class class wbr 4685  cmpt 4762  dom cdm 5143  ran crn 5144  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by:  ordtbas2  21043  ordtbas  21044  ordttopon  21045  ordtopn1  21046  ordtopn2  21047  ordtrest2  21056  ordthmeolem  21652  ordtprsuni  30093
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