Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mpstssv Structured version   Visualization version   GIF version

Theorem mpstssv 30690
 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 2610 . . 3 (mDV‘𝑇) = (mDV‘𝑇)
2 eqid 2610 . . 3 (mEx‘𝑇) = (mEx‘𝑇)
3 mpstssv.p . . 3 𝑃 = (mPreSt‘𝑇)
41, 2, 3mpstval 30686 . 2 𝑃 = (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇))
5 xpss 5149 . . 3 ({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) ⊆ (V × V)
6 ssv 3588 . . 3 (mEx‘𝑇) ⊆ V
7 xpss12 5148 . . 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 704 . 2 (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇)) ⊆ ((V × V) × V)
94, 8eqsstri 3598 1 𝑃 ⊆ ((V × V) × V)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108   × cxp 5036  ◡ccnv 5037  ‘cfv 5804  Fincfn 7841  mExcmex 30618  mDVcmdv 30619  mPreStcmpst 30624 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-mpst 30644 This theorem is referenced by:  mpst123  30691  mpstrcl  30692  msrrcl  30694  elmpps  30724
 Copyright terms: Public domain W3C validator