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Theorem ordtuni 21790
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 7605 . . . . . 6 (𝑅𝑉 → dom 𝑅 ∈ V)
31, 2eqeltrid 2915 . . . . 5 (𝑅𝑉𝑋 ∈ V)
4 unisng 4845 . . . . 5 (𝑋 ∈ V → {𝑋} = 𝑋)
53, 4syl 17 . . . 4 (𝑅𝑉 {𝑋} = 𝑋)
65uneq1d 4136 . . 3 (𝑅𝑉 → ( {𝑋} ∪ (𝐴𝐵)) = (𝑋 (𝐴𝐵)))
7 ordtval.2 . . . . . . 7 𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
8 ssrab2 4054 . . . . . . . . . 10 {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
93adantr 483 . . . . . . . . . . 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 260 . . . . . . . . 9 ((𝑅𝑉𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋)
1312fmpttd 6872 . . . . . . . 8 (𝑅𝑉 → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
1413frnd 6514 . . . . . . 7 (𝑅𝑉 → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
157, 14eqsstrid 4013 . . . . . 6 (𝑅𝑉𝐴 ⊆ 𝒫 𝑋)
16 ordtval.3 . . . . . . 7 𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
17 ssrab2 4054 . . . . . . . . . 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 260 . . . . . . . . 9 ((𝑅𝑉𝑥𝑋) → {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ 𝒫 𝑋)
2120fmpttd 6872 . . . . . . . 8 (𝑅𝑉 → (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}):𝑋⟶𝒫 𝑋)
2221frnd 6514 . . . . . . 7 (𝑅𝑉 → ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⊆ 𝒫 𝑋)
2316, 22eqsstrid 4013 . . . . . 6 (𝑅𝑉𝐵 ⊆ 𝒫 𝑋)
2415, 23unssd 4160 . . . . 5 (𝑅𝑉 → (𝐴𝐵) ⊆ 𝒫 𝑋)
25 sspwuni 5013 . . . . 5 ((𝐴𝐵) ⊆ 𝒫 𝑋 (𝐴𝐵) ⊆ 𝑋)
2624, 25sylib 220 . . . 4 (𝑅𝑉 (𝐴𝐵) ⊆ 𝑋)
27 ssequn2 4157 . . . 4 ( (𝐴𝐵) ⊆ 𝑋 ↔ (𝑋 (𝐴𝐵)) = 𝑋)
2826, 27sylib 220 . . 3 (𝑅𝑉 → (𝑋 (𝐴𝐵)) = 𝑋)
296, 28eqtr2d 2855 . 2 (𝑅𝑉𝑋 = ( {𝑋} ∪ (𝐴𝐵)))
30 uniun 4849 . 2 ({𝑋} ∪ (𝐴𝐵)) = ( {𝑋} ∪ (𝐴𝐵))
3129, 30syl6eqr 2872 1 (𝑅𝑉𝑋 = ({𝑋} ∪ (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1531  wcel 2108  {crab 3140  Vcvv 3493  cun 3932  wss 3934  𝒫 cpw 4537  {csn 4559   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 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  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  21791  ordtbas  21792  ordttopon  21793  ordtopn1  21794  ordtopn2  21795  ordtrest2  21804  ordthmeolem  22401  ordtprsuni  31155
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