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Theorem ordtuni 21196
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 7262 . . . . . 6 (𝑅𝑉 → dom 𝑅 ∈ V)
31, 2syl5eqel 2843 . . . . 5 (𝑅𝑉𝑋 ∈ V)
4 unisng 4604 . . . . 5 (𝑋 ∈ V → {𝑋} = 𝑋)
53, 4syl 17 . . . 4 (𝑅𝑉 {𝑋} = 𝑋)
65uneq1d 3909 . . 3 (𝑅𝑉 → ( {𝑋} ∪ (𝐴𝐵)) = (𝑋 (𝐴𝐵)))
7 ordtval.2 . . . . . . 7 𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
8 ssrab2 3828 . . . . . . . . . 10 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
93adantr 472 . . . . . . . . . . 11 ((𝑅𝑉𝑥𝑋) → 𝑋 ∈ V)
10 elpw2g 4976 . . . . . . . . . . 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 2760 . . . . . . . . 9 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
1412, 13fmptd 6548 . . . . . . . 8 (𝑅𝑉 → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
15 frn 6214 . . . . . . . 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 3790 . . . . . 6 (𝑅𝑉𝐴 ⊆ 𝒫 𝑋)
18 ordtval.3 . . . . . . 7 𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
19 ssrab2 3828 . . . . . . . . . 10 {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋
20 elpw2g 4976 . . . . . . . . . . 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 2760 . . . . . . . . 9 (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
2422, 23fmptd 6548 . . . . . . . 8 (𝑅𝑉 → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}):𝑋⟶𝒫 𝑋)
25 frn 6214 . . . . . . . 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 3790 . . . . . 6 (𝑅𝑉𝐵 ⊆ 𝒫 𝑋)
2817, 27unssd 3932 . . . . 5 (𝑅𝑉 → (𝐴𝐵) ⊆ 𝒫 𝑋)
29 sspwuni 4763 . . . . 5 ((𝐴𝐵) ⊆ 𝒫 𝑋 (𝐴𝐵) ⊆ 𝑋)
3028, 29sylib 208 . . . 4 (𝑅𝑉 (𝐴𝐵) ⊆ 𝑋)
31 ssequn2 3929 . . . 4 ( (𝐴𝐵) ⊆ 𝑋 ↔ (𝑋 (𝐴𝐵)) = 𝑋)
3230, 31sylib 208 . . 3 (𝑅𝑉 → (𝑋 (𝐴𝐵)) = 𝑋)
336, 32eqtr2d 2795 . 2 (𝑅𝑉𝑋 = ( {𝑋} ∪ (𝐴𝐵)))
34 uniun 4608 . 2 ({𝑋} ∪ (𝐴𝐵)) = ( {𝑋} ∪ (𝐴𝐵))
3533, 34syl6eqr 2812 1 (𝑅𝑉𝑋 = ({𝑋} ∪ (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  cun 3713  wss 3715  𝒫 cpw 4302  {csn 4321   cuni 4588   class class class wbr 4804  cmpt 4881  dom cdm 5266  ran crn 5267  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057
This theorem is referenced by:  ordtbas2  21197  ordtbas  21198  ordttopon  21199  ordtopn1  21200  ordtopn2  21201  ordtrest2  21210  ordthmeolem  21806  ordtprsuni  30274
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