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Theorem mpstssv 31562
Description: A pre-statement is an ordered triple. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
mpstssv.p 𝑃 = (mPreSt‘𝑇)
Assertion
Ref Expression
mpstssv 𝑃 ⊆ ((V × V) × V)

Proof of Theorem mpstssv
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . 3 (mDV‘𝑇) = (mDV‘𝑇)
2 eqid 2651 . . 3 (mEx‘𝑇) = (mEx‘𝑇)
3 mpstssv.p . . 3 𝑃 = (mPreSt‘𝑇)
41, 2, 3mpstval 31558 . 2 𝑃 = (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇))
5 xpss 5159 . . 3 ({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) ⊆ (V × V)
6 ssv 3658 . . 3 (mEx‘𝑇) ⊆ V
7 xpss12 5158 . . 3 ((({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) ⊆ (V × V) ∧ (mEx‘𝑇) ⊆ V) → (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇)) ⊆ ((V × V) × V))
85, 6, 7mp2an 708 . 2 (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇)) ⊆ ((V × V) × V)
94, 8eqsstri 3668 1 𝑃 ⊆ ((V × V) × V)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  {crab 2945  Vcvv 3231  cin 3606  wss 3607  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:  mpst123  31563  mpstrcl  31564  msrrcl  31566  elmpps  31596
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