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Theorem mpstval 34216
Description: A pre-statement is an ordered triple, whose first member is a symmetric set of disjoint variable conditions, whose second member is a finite set of expressions, and whose third member is an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mpstval.v 𝑉 = (mDV‘𝑇)
mpstval.e 𝐸 = (mEx‘𝑇)
mpstval.p 𝑃 = (mPreSt‘𝑇)
Assertion
Ref Expression
mpstval 𝑃 = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
Distinct variable groups:   𝑇,𝑑   𝑉,𝑑
Allowed substitution hints:   𝑃(𝑑)   𝐸(𝑑)

Proof of Theorem mpstval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 mpstval.p . 2 𝑃 = (mPreSt‘𝑇)
2 fveq2 6847 . . . . . . . . 9 (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
3 mpstval.v . . . . . . . . 9 𝑉 = (mDV‘𝑇)
42, 3eqtr4di 2789 . . . . . . . 8 (𝑡 = 𝑇 → (mDV‘𝑡) = 𝑉)
54pweqd 4582 . . . . . . 7 (𝑡 = 𝑇 → 𝒫 (mDV‘𝑡) = 𝒫 𝑉)
65rabeqdv 3420 . . . . . 6 (𝑡 = 𝑇 → {𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ 𝑑 = 𝑑} = {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑})
7 fveq2 6847 . . . . . . . . 9 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
8 mpstval.e . . . . . . . . 9 𝐸 = (mEx‘𝑇)
97, 8eqtr4di 2789 . . . . . . . 8 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
109pweqd 4582 . . . . . . 7 (𝑡 = 𝑇 → 𝒫 (mEx‘𝑡) = 𝒫 𝐸)
1110ineq1d 4176 . . . . . 6 (𝑡 = 𝑇 → (𝒫 (mEx‘𝑡) ∩ Fin) = (𝒫 𝐸 ∩ Fin))
126, 11xpeq12d 5669 . . . . 5 (𝑡 = 𝑇 → ({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) = ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)))
1312, 9xpeq12d 5669 . . . 4 (𝑡 = 𝑇 → (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
14 df-mpst 34174 . . . 4 mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
153fvexi 6861 . . . . . . . 8 𝑉 ∈ V
1615pwex 5340 . . . . . . 7 𝒫 𝑉 ∈ V
1716rabex 5294 . . . . . 6 {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ∈ V
188fvexi 6861 . . . . . . . 8 𝐸 ∈ V
1918pwex 5340 . . . . . . 7 𝒫 𝐸 ∈ V
2019inex1 5279 . . . . . 6 (𝒫 𝐸 ∩ Fin) ∈ V
2117, 20xpex 7692 . . . . 5 ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∈ V
2221, 18xpex 7692 . . . 4 (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ∈ V
2313, 14, 22fvmpt 6953 . . 3 (𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
24 xp0 6115 . . . . 5 (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅) = ∅
2524eqcomi 2740 . . . 4 ∅ = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅)
26 fvprc 6839 . . . 4 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
27 fvprc 6839 . . . . . 6 𝑇 ∈ V → (mEx‘𝑇) = ∅)
288, 27eqtrid 2783 . . . . 5 𝑇 ∈ V → 𝐸 = ∅)
2928xpeq2d 5668 . . . 4 𝑇 ∈ V → (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅))
3025, 26, 293eqtr4a 2797 . . 3 𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
3123, 30pm2.61i 182 . 2 (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
321, 31eqtri 2759 1 𝑃 = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wcel 2106  {crab 3405  Vcvv 3446  cin 3912  c0 4287  𝒫 cpw 4565   × cxp 5636  ccnv 5637  cfv 6501  Fincfn 8890  mExcmex 34148  mDVcmdv 34149  mPreStcmpst 34154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-mpst 34174
This theorem is referenced by:  elmpst  34217  mpstssv  34220
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