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Theorem ordtuni 20899
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 7045 . . . . . 6 (𝑅𝑉 → dom 𝑅 ∈ V)
31, 2syl5eqel 2708 . . . . 5 (𝑅𝑉𝑋 ∈ V)
4 unisng 4423 . . . . 5 (𝑋 ∈ V → {𝑋} = 𝑋)
53, 4syl 17 . . . 4 (𝑅𝑉 {𝑋} = 𝑋)
65uneq1d 3749 . . 3 (𝑅𝑉 → ( {𝑋} ∪ (𝐴𝐵)) = (𝑋 (𝐴𝐵)))
7 ordtval.2 . . . . . . 7 𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
8 ssrab2 3671 . . . . . . . . . 10 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
93adantr 481 . . . . . . . . . . 11 ((𝑅𝑉𝑥𝑋) → 𝑋 ∈ V)
10 elpw2g 4792 . . . . . . . . . . 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 2626 . . . . . . . . 9 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
1412, 13fmptd 6341 . . . . . . . 8 (𝑅𝑉 → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
15 frn 6012 . . . . . . . 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 3633 . . . . . 6 (𝑅𝑉𝐴 ⊆ 𝒫 𝑋)
18 ordtval.3 . . . . . . 7 𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
19 ssrab2 3671 . . . . . . . . . 10 {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋
20 elpw2g 4792 . . . . . . . . . . 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 2626 . . . . . . . . 9 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
2422, 23fmptd 6341 . . . . . . . 8 (𝑅𝑉 → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}):𝑋⟶𝒫 𝑋)
25 frn 6012 . . . . . . . 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 3633 . . . . . 6 (𝑅𝑉𝐵 ⊆ 𝒫 𝑋)
2817, 27unssd 3772 . . . . 5 (𝑅𝑉 → (𝐴𝐵) ⊆ 𝒫 𝑋)
29 sspwuni 4582 . . . . 5 ((𝐴𝐵) ⊆ 𝒫 𝑋 (𝐴𝐵) ⊆ 𝑋)
3028, 29sylib 208 . . . 4 (𝑅𝑉 (𝐴𝐵) ⊆ 𝑋)
31 ssequn2 3769 . . . 4 ( (𝐴𝐵) ⊆ 𝑋 ↔ (𝑋 (𝐴𝐵)) = 𝑋)
3230, 31sylib 208 . . 3 (𝑅𝑉 → (𝑋 (𝐴𝐵)) = 𝑋)
336, 32eqtr2d 2661 . 2 (𝑅𝑉𝑋 = ( {𝑋} ∪ (𝐴𝐵)))
34 uniun 4427 . 2 ({𝑋} ∪ (𝐴𝐵)) = ( {𝑋} ∪ (𝐴𝐵))
3533, 34syl6eqr 2678 1 (𝑅𝑉𝑋 = ({𝑋} ∪ (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  {crab 2916  Vcvv 3191  cun 3558  wss 3560  𝒫 cpw 4135  {csn 4153   cuni 4407   class class class wbr 4618  cmpt 4678  dom cdm 5079  ran crn 5080  wf 5846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858
This theorem is referenced by:  ordtbas2  20900  ordtbas  20901  ordttopon  20902  ordtopn1  20903  ordtopn2  20904  ordtrest2  20913  ordthmeolem  21509  ordtprsuni  29739
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