Theorem mpstval 30686

Description: A pre-statement is an ordered triple, whose first member is a symmetric set of dv 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 = (mDV‘𝑇)
mpstval.e	⊢ 𝐸 = (mEx‘𝑇)
mpstval.p	⊢ 𝑃 = (mPreSt‘𝑇)

Assertion

Ref	Expression
mpstval	⊢ 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)

Distinct variable groups:   𝑇,𝑑   𝑉,𝑑

Allowed substitution hints:   𝑃(𝑑)   𝐸(𝑑)

Proof of Theorem mpstval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpstval.p	. 2 ⊢ 𝑃 = (mPreSt‘𝑇)
2		fveq2 6103	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
3		mpstval.v	. . . . . . . . 9 ⊢ 𝑉 = (mDV‘𝑇)
4	2, 3	syl6eqr 2662	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = 𝑉)
5	4	pweqd 4113	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝒫 (mDV‘𝑡) = 𝒫 𝑉)
6	5	rabeqdv 3167	. . . . . 6 ⊢ (𝑡 = 𝑇 → {𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} = {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑})
7		fveq2 6103	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
8		mpstval.e	. . . . . . . . 9 ⊢ 𝐸 = (mEx‘𝑇)
9	7, 8	syl6eqr 2662	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
10	9	pweqd 4113	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝒫 (mEx‘𝑡) = 𝒫 𝐸)
11	10	ineq1d 3775	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝒫 (mEx‘𝑡) ∩ Fin) = (𝒫 𝐸 ∩ Fin))
12	6, 11	xpeq12d 5064	. . . . 5 ⊢ (𝑡 = 𝑇 → ({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) = ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)))
13	12, 9	xpeq12d 5064	. . . 4 ⊢ (𝑡 = 𝑇 → (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
14		df-mpst 30644	. . . 4 ⊢ mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
15		fvex 6113	. . . . . . . . 9 ⊢ (mDV‘𝑇) ∈ V
16	3, 15	eqeltri 2684	. . . . . . . 8 ⊢ 𝑉 ∈ V
17	16	pwex 4774	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
18	17	rabex 4740	. . . . . 6 ⊢ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ∈ V
19		fvex 6113	. . . . . . . . 9 ⊢ (mEx‘𝑇) ∈ V
20	8, 19	eqeltri 2684	. . . . . . . 8 ⊢ 𝐸 ∈ V
21	20	pwex 4774	. . . . . . 7 ⊢ 𝒫 𝐸 ∈ V
22	21	inex1 4727	. . . . . 6 ⊢ (𝒫 𝐸 ∩ Fin) ∈ V
23	18, 22	xpex 6860	. . . . 5 ⊢ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∈ V
24	23, 20	xpex 6860	. . . 4 ⊢ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ∈ V
25	13, 14, 24	fvmpt 6191	. . 3 ⊢ (𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
26		xp0 5471	. . . . 5 ⊢ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅) = ∅
27	26	eqcomi 2619	. . . 4 ⊢ ∅ = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅)
28		fvprc 6097	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
29		fvprc 6097	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
30	8, 29	syl5eq 2656	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐸 = ∅)
31	30	xpeq2d 5063	. . . 4 ⊢ (¬ 𝑇 ∈ V → (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅))
32	27, 28, 31	3eqtr4a 2670	. . 3 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
33	25, 32	pm2.61i 175	. 2 ⊢ (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
34	1, 33	eqtri 2632	1 ⊢ 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∩ cin 3539  ∅c0 3874  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:  elmpst  30687  mpstssv  30690