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Theorem mpstssv 32804
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 2824 . . 3 (mDV‘𝑇) = (mDV‘𝑇)
2 eqid 2824 . . 3 (mEx‘𝑇) = (mEx‘𝑇)
3 mpstssv.p . . 3 𝑃 = (mPreSt‘𝑇)
41, 2, 3mpstval 32800 . 2 𝑃 = (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇))
5 xpss 5552 . . 3 ({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) ⊆ (V × V)
6 ssv 3975 . . 3 (mEx‘𝑇) ⊆ V
7 xpss12 5551 . . 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 691 . 2 (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇)) ⊆ ((V × V) × V)
94, 8eqsstri 3985 1 𝑃 ⊆ ((V × V) × V)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  {crab 3136  Vcvv 3479  cin 3917  wss 3918  𝒫 cpw 4520   × cxp 5534  ccnv 5535  cfv 6336  Fincfn 8492  mExcmex 32732  mDVcmdv 32733  mPreStcmpst 32738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6295  df-fun 6338  df-fv 6344  df-mpst 32758
This theorem is referenced by:  mpst123  32805  mpstrcl  32806  msrrcl  32808  elmpps  32838
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