Theorem mpstval 36269

Description: A pre-statement is an ordered triple, whose first member is a symmetric set of disjoint variable 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 6877	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
3		mpstval.v	. . . . . . . . 9 ⊢ 𝑉 = (mDV‘𝑇)
4	2, 3	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = 𝑉)
5	4	pweqd 4574	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝒫 (mDV‘𝑡) = 𝒫 𝑉)
6	5	rabeqdv 3428	. . . . . 6 ⊢ (𝑡 = 𝑇 → {𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} = {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑})
7		fveq2 6877	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
8		mpstval.e	. . . . . . . . 9 ⊢ 𝐸 = (mEx‘𝑇)
9	7, 8	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
10	9	pweqd 4574	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝒫 (mEx‘𝑡) = 𝒫 𝐸)
11	10	ineq1d 4165	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝒫 (mEx‘𝑡) ∩ Fin) = (𝒫 𝐸 ∩ Fin))
12	6, 11	xpeq12d 5682	. . . . 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 5682	. . . 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 36227	. . . 4 ⊢ mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
15	3	fvexi 6891	. . . . . . . 8 ⊢ 𝑉 ∈ V
16	15	pwex 5342	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
17	16	rabex 5300	. . . . . 6 ⊢ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ∈ V
18	8	fvexi 6891	. . . . . . . 8 ⊢ 𝐸 ∈ V
19	18	pwex 5342	. . . . . . 7 ⊢ 𝒫 𝐸 ∈ V
20	19	inex1 5277	. . . . . 6 ⊢ (𝒫 𝐸 ∩ Fin) ∈ V
21	17, 20	xpex 7756	. . . . 5 ⊢ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∈ V
22	21, 18	xpex 7756	. . . 4 ⊢ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ∈ V
23	13, 14, 22	fvmpt 6985	. . 3 ⊢ (𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
24		xp0 5751	. . . . 5 ⊢ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅) = ∅
25	24	eqcomi 2770	. . . 4 ⊢ ∅ = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅)
26		fvprc 6869	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
27		fvprc 6869	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
28	8, 27	eqtrid 2808	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐸 = ∅)
29	28	xpeq2d 5681	. . . 4 ⊢ (¬ 𝑇 ∈ V → (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅))
30	25, 26, 29	3eqtr4a 2822	. . 3 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
31	23, 30	pm2.61i 184	. 2 ⊢ (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
32	1, 31	eqtri 2784	1 ⊢ 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  {crab 3413  Vcvv 3451   ∩ cin 3898  ∅c0 4279  𝒫 cpw 4557   × cxp 5649  ^◡ccnv 5650  ‘cfv 6531  Fincfn 8957  mExcmex 36201  mDVcmdv 36202  mPreStcmpst 36207

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-mpst 36227

This theorem is used by:  elmpst  36270  mpstssv  36273