Theorem mpstssv 32788

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.

Step	Hyp	Ref	Expression
1		eqid 2823	. . 3 ⊢ (mDV‘𝑇) = (mDV‘𝑇)
2		eqid 2823	. . 3 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
3		mpstssv.p	. . 3 ⊢ 𝑃 = (mPreSt‘𝑇)
4	1, 2, 3	mpstval 32784	. 2 ⊢ 𝑃 = (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇))
5		xpss 5573	. . 3 ⊢ ({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) ⊆ (V × V)
6		ssv 3993	. . 3 ⊢ (mEx‘𝑇) ⊆ V
7		xpss12 5572	. . 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))
8	5, 6, 7	mp2an 690	. 2 ⊢ (({𝑑 ∈ 𝒫 (mDV‘𝑇) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑇) ∩ Fin)) × (mEx‘𝑇)) ⊆ ((V × V) × V)
9	4, 8	eqsstri 4003	1 ⊢ 𝑃 ⊆ ((V × V) × V)

Syntax hints:   = wceq 1537  {crab 3144  Vcvv 3496   ∩ cin 3937   ⊆ wss 3938  𝒫 cpw 4541   × cxp 5555  ^◡ccnv 5556  ‘cfv 6357  Fincfn 8511  mExcmex 32716  mDVcmdv 32717  mPreStcmpst 32722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-mpst 32742

This theorem is referenced by:  mpst123  32789  mpstrcl  32790  msrrcl  32792  elmpps  32822