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Theorem mpstval 33210
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 6717 . . . . . . . . 9 (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
3 mpstval.v . . . . . . . . 9 𝑉 = (mDV‘𝑇)
42, 3eqtr4di 2796 . . . . . . . 8 (𝑡 = 𝑇 → (mDV‘𝑡) = 𝑉)
54pweqd 4532 . . . . . . 7 (𝑡 = 𝑇 → 𝒫 (mDV‘𝑡) = 𝒫 𝑉)
65rabeqdv 3395 . . . . . 6 (𝑡 = 𝑇 → {𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ 𝑑 = 𝑑} = {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑})
7 fveq2 6717 . . . . . . . . 9 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
8 mpstval.e . . . . . . . . 9 𝐸 = (mEx‘𝑇)
97, 8eqtr4di 2796 . . . . . . . 8 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
109pweqd 4532 . . . . . . 7 (𝑡 = 𝑇 → 𝒫 (mEx‘𝑡) = 𝒫 𝐸)
1110ineq1d 4126 . . . . . 6 (𝑡 = 𝑇 → (𝒫 (mEx‘𝑡) ∩ Fin) = (𝒫 𝐸 ∩ Fin))
126, 11xpeq12d 5582 . . . . 5 (𝑡 = 𝑇 → ({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) = ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)))
1312, 9xpeq12d 5582 . . . 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 33168 . . . 4 mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
153fvexi 6731 . . . . . . . 8 𝑉 ∈ V
1615pwex 5273 . . . . . . 7 𝒫 𝑉 ∈ V
1716rabex 5225 . . . . . 6 {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ∈ V
188fvexi 6731 . . . . . . . 8 𝐸 ∈ V
1918pwex 5273 . . . . . . 7 𝒫 𝐸 ∈ V
2019inex1 5210 . . . . . 6 (𝒫 𝐸 ∩ Fin) ∈ V
2117, 20xpex 7538 . . . . 5 ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∈ V
2221, 18xpex 7538 . . . 4 (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ∈ V
2313, 14, 22fvmpt 6818 . . 3 (𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
24 xp0 6021 . . . . 5 (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅) = ∅
2524eqcomi 2746 . . . 4 ∅ = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅)
26 fvprc 6709 . . . 4 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
27 fvprc 6709 . . . . . 6 𝑇 ∈ V → (mEx‘𝑇) = ∅)
288, 27syl5eq 2790 . . . . 5 𝑇 ∈ V → 𝐸 = ∅)
2928xpeq2d 5581 . . . 4 𝑇 ∈ V → (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅))
3025, 26, 293eqtr4a 2804 . . 3 𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
3123, 30pm2.61i 185 . 2 (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
321, 31eqtri 2765 1 𝑃 = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1543  wcel 2110  {crab 3065  Vcvv 3408  cin 3865  c0 4237  𝒫 cpw 4513   × cxp 5549  ccnv 5550  cfv 6380  Fincfn 8626  mExcmex 33142  mDVcmdv 33143  mPreStcmpst 33148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-mpst 33168
This theorem is referenced by:  elmpst  33211  mpstssv  33214
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