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Theorem mpstval 31558
Description: A pre-statement is an ordered triple, whose first member is a symmetric set of dv 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 6229 . . . . . . . . 9 (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
3 mpstval.v . . . . . . . . 9 𝑉 = (mDV‘𝑇)
42, 3syl6eqr 2703 . . . . . . . 8 (𝑡 = 𝑇 → (mDV‘𝑡) = 𝑉)
54pweqd 4196 . . . . . . 7 (𝑡 = 𝑇 → 𝒫 (mDV‘𝑡) = 𝒫 𝑉)
65rabeqdv 3225 . . . . . 6 (𝑡 = 𝑇 → {𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ 𝑑 = 𝑑} = {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑})
7 fveq2 6229 . . . . . . . . 9 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
8 mpstval.e . . . . . . . . 9 𝐸 = (mEx‘𝑇)
97, 8syl6eqr 2703 . . . . . . . 8 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
109pweqd 4196 . . . . . . 7 (𝑡 = 𝑇 → 𝒫 (mEx‘𝑡) = 𝒫 𝐸)
1110ineq1d 3846 . . . . . 6 (𝑡 = 𝑇 → (𝒫 (mEx‘𝑡) ∩ Fin) = (𝒫 𝐸 ∩ Fin))
126, 11xpeq12d 5174 . . . . 5 (𝑡 = 𝑇 → ({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) = ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)))
1312, 9xpeq12d 5174 . . . 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 31516 . . . 4 mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
15 fvex 6239 . . . . . . . . 9 (mDV‘𝑇) ∈ V
163, 15eqeltri 2726 . . . . . . . 8 𝑉 ∈ V
1716pwex 4878 . . . . . . 7 𝒫 𝑉 ∈ V
1817rabex 4845 . . . . . 6 {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ∈ V
19 fvex 6239 . . . . . . . . 9 (mEx‘𝑇) ∈ V
208, 19eqeltri 2726 . . . . . . . 8 𝐸 ∈ V
2120pwex 4878 . . . . . . 7 𝒫 𝐸 ∈ V
2221inex1 4832 . . . . . 6 (𝒫 𝐸 ∩ Fin) ∈ V
2318, 22xpex 7004 . . . . 5 ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∈ V
2423, 20xpex 7004 . . . 4 (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ∈ V
2513, 14, 24fvmpt 6321 . . 3 (𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
26 xp0 5587 . . . . 5 (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅) = ∅
2726eqcomi 2660 . . . 4 ∅ = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅)
28 fvprc 6223 . . . 4 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
29 fvprc 6223 . . . . . 6 𝑇 ∈ V → (mEx‘𝑇) = ∅)
308, 29syl5eq 2697 . . . . 5 𝑇 ∈ V → 𝐸 = ∅)
3130xpeq2d 5173 . . . 4 𝑇 ∈ V → (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅))
3227, 28, 313eqtr4a 2711 . . 3 𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
3325, 32pm2.61i 176 . 2 (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
341, 33eqtri 2673 1 𝑃 = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  cin 3606  c0 3948  𝒫 cpw 4191   × cxp 5141  ccnv 5142  cfv 5926  Fincfn 7997  mExcmex 31490  mDVcmdv 31491  mPreStcmpst 31496
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-pow 4873  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-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-iota 5889  df-fun 5928  df-fv 5934  df-mpst 31516
This theorem is referenced by:  elmpst  31559  mpstssv  31562
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