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Theorem mpstval 34186
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 6843 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (mDVβ€˜π‘‘) = (mDVβ€˜π‘‡))
3 mpstval.v . . . . . . . . 9 𝑉 = (mDVβ€˜π‘‡)
42, 3eqtr4di 2791 . . . . . . . 8 (𝑑 = 𝑇 β†’ (mDVβ€˜π‘‘) = 𝑉)
54pweqd 4578 . . . . . . 7 (𝑑 = 𝑇 β†’ 𝒫 (mDVβ€˜π‘‘) = 𝒫 𝑉)
65rabeqdv 3421 . . . . . 6 (𝑑 = 𝑇 β†’ {𝑑 ∈ 𝒫 (mDVβ€˜π‘‘) ∣ ◑𝑑 = 𝑑} = {𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑})
7 fveq2 6843 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = (mExβ€˜π‘‡))
8 mpstval.e . . . . . . . . 9 𝐸 = (mExβ€˜π‘‡)
97, 8eqtr4di 2791 . . . . . . . 8 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = 𝐸)
109pweqd 4578 . . . . . . 7 (𝑑 = 𝑇 β†’ 𝒫 (mExβ€˜π‘‘) = 𝒫 𝐸)
1110ineq1d 4172 . . . . . 6 (𝑑 = 𝑇 β†’ (𝒫 (mExβ€˜π‘‘) ∩ Fin) = (𝒫 𝐸 ∩ Fin))
126, 11xpeq12d 5665 . . . . 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 5665 . . . 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 34144 . . . 4 mPreSt = (𝑑 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDVβ€˜π‘‘) ∣ ◑𝑑 = 𝑑} Γ— (𝒫 (mExβ€˜π‘‘) ∩ Fin)) Γ— (mExβ€˜π‘‘)))
153fvexi 6857 . . . . . . . 8 𝑉 ∈ V
1615pwex 5336 . . . . . . 7 𝒫 𝑉 ∈ V
1716rabex 5290 . . . . . 6 {𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} ∈ V
188fvexi 6857 . . . . . . . 8 𝐸 ∈ V
1918pwex 5336 . . . . . . 7 𝒫 𝐸 ∈ V
2019inex1 5275 . . . . . 6 (𝒫 𝐸 ∩ Fin) ∈ V
2117, 20xpex 7688 . . . . 5 ({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) ∈ V
2221, 18xpex 7688 . . . 4 (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸) ∈ V
2313, 14, 22fvmpt 6949 . . 3 (𝑇 ∈ V β†’ (mPreStβ€˜π‘‡) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸))
24 xp0 6111 . . . . 5 (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— βˆ…) = βˆ…
2524eqcomi 2742 . . . 4 βˆ… = (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— βˆ…)
26 fvprc 6835 . . . 4 (Β¬ 𝑇 ∈ V β†’ (mPreStβ€˜π‘‡) = βˆ…)
27 fvprc 6835 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mExβ€˜π‘‡) = βˆ…)
288, 27eqtrid 2785 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝐸 = βˆ…)
2928xpeq2d 5664 . . . 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 2799 . . 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 2761 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 1542   ∈ wcel 2107  {crab 3406  Vcvv 3444   ∩ cin 3910  βˆ…c0 4283  π’« cpw 4561   Γ— cxp 5632  β—‘ccnv 5633  β€˜cfv 6497  Fincfn 8886  mExcmex 34118  mDVcmdv 34119  mPreStcmpst 34124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-mpst 34144
This theorem is referenced by:  elmpst  34187  mpstssv  34190
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