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Theorem mpstval 35081
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 = (mDVβ€˜π‘‡)
mpstval.e 𝐸 = (mExβ€˜π‘‡)
mpstval.p 𝑃 = (mPreStβ€˜π‘‡)
Assertion
Ref Expression
mpstval 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸)
Distinct variable groups:   𝑇,𝑑   𝑉,𝑑
Allowed substitution hints:   𝑃(𝑑)   𝐸(𝑑)

Proof of Theorem mpstval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 mpstval.p . 2 𝑃 = (mPreStβ€˜π‘‡)
2 fveq2 6891 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (mDVβ€˜π‘‘) = (mDVβ€˜π‘‡))
3 mpstval.v . . . . . . . . 9 𝑉 = (mDVβ€˜π‘‡)
42, 3eqtr4di 2785 . . . . . . . 8 (𝑑 = 𝑇 β†’ (mDVβ€˜π‘‘) = 𝑉)
54pweqd 4615 . . . . . . 7 (𝑑 = 𝑇 β†’ 𝒫 (mDVβ€˜π‘‘) = 𝒫 𝑉)
65rabeqdv 3442 . . . . . 6 (𝑑 = 𝑇 β†’ {𝑑 ∈ 𝒫 (mDVβ€˜π‘‘) ∣ ◑𝑑 = 𝑑} = {𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑})
7 fveq2 6891 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = (mExβ€˜π‘‡))
8 mpstval.e . . . . . . . . 9 𝐸 = (mExβ€˜π‘‡)
97, 8eqtr4di 2785 . . . . . . . 8 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = 𝐸)
109pweqd 4615 . . . . . . 7 (𝑑 = 𝑇 β†’ 𝒫 (mExβ€˜π‘‘) = 𝒫 𝐸)
1110ineq1d 4207 . . . . . 6 (𝑑 = 𝑇 β†’ (𝒫 (mExβ€˜π‘‘) ∩ Fin) = (𝒫 𝐸 ∩ Fin))
126, 11xpeq12d 5703 . . . . 5 (𝑑 = 𝑇 β†’ ({𝑑 ∈ 𝒫 (mDVβ€˜π‘‘) ∣ ◑𝑑 = 𝑑} Γ— (𝒫 (mExβ€˜π‘‘) ∩ Fin)) = ({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)))
1312, 9xpeq12d 5703 . . . 4 (𝑑 = 𝑇 β†’ (({𝑑 ∈ 𝒫 (mDVβ€˜π‘‘) ∣ ◑𝑑 = 𝑑} Γ— (𝒫 (mExβ€˜π‘‘) ∩ Fin)) Γ— (mExβ€˜π‘‘)) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸))
14 df-mpst 35039 . . . 4 mPreSt = (𝑑 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDVβ€˜π‘‘) ∣ ◑𝑑 = 𝑑} Γ— (𝒫 (mExβ€˜π‘‘) ∩ Fin)) Γ— (mExβ€˜π‘‘)))
153fvexi 6905 . . . . . . . 8 𝑉 ∈ V
1615pwex 5374 . . . . . . 7 𝒫 𝑉 ∈ V
1716rabex 5328 . . . . . 6 {𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} ∈ V
188fvexi 6905 . . . . . . . 8 𝐸 ∈ V
1918pwex 5374 . . . . . . 7 𝒫 𝐸 ∈ V
2019inex1 5311 . . . . . 6 (𝒫 𝐸 ∩ Fin) ∈ V
2117, 20xpex 7749 . . . . 5 ({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) ∈ V
2221, 18xpex 7749 . . . 4 (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸) ∈ V
2313, 14, 22fvmpt 6999 . . 3 (𝑇 ∈ V β†’ (mPreStβ€˜π‘‡) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸))
24 xp0 6156 . . . . 5 (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— βˆ…) = βˆ…
2524eqcomi 2736 . . . 4 βˆ… = (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— βˆ…)
26 fvprc 6883 . . . 4 (Β¬ 𝑇 ∈ V β†’ (mPreStβ€˜π‘‡) = βˆ…)
27 fvprc 6883 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mExβ€˜π‘‡) = βˆ…)
288, 27eqtrid 2779 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝐸 = βˆ…)
2928xpeq2d 5702 . . . 4 (Β¬ 𝑇 ∈ V β†’ (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— βˆ…))
3025, 26, 293eqtr4a 2793 . . 3 (Β¬ 𝑇 ∈ V β†’ (mPreStβ€˜π‘‡) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸))
3123, 30pm2.61i 182 . 2 (mPreStβ€˜π‘‡) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸)
321, 31eqtri 2755 1 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1534   ∈ wcel 2099  {crab 3427  Vcvv 3469   ∩ cin 3943  βˆ…c0 4318  π’« cpw 4598   Γ— cxp 5670  β—‘ccnv 5671  β€˜cfv 6542  Fincfn 8955  mExcmex 35013  mDVcmdv 35014  mPreStcmpst 35019
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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-mpst 35039
This theorem is referenced by:  elmpst  35082  mpstssv  35085
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