Theorem elmpst 36049

Description: Property of being a pre-statement. (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
elmpst	⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ 𝐸))

Proof of Theorem elmpst

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelxp 5702	. . 3 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ↔ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∧ 𝐴 ∈ 𝐸))
2		opelxp 5702	. . . . 5 ⊢ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)))
3		cnveq 5864	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ◡𝑑 = ◡𝐷)
4		id 23	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → 𝑑 = 𝐷)
5	3, 4	eqeq12d 2782	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (◡𝑑 = 𝑑 ↔ ◡𝐷 = 𝐷))
6	5	elrab 3653	. . . . . . 7 ⊢ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ↔ (𝐷 ∈ 𝒫 𝑉 ∧ ◡𝐷 = 𝐷))
7		mpstval.v	. . . . . . . . . 10 ⊢ 𝑉 = (mDV‘𝑇)
8	7	fvexi 6902	. . . . . . . . 9 ⊢ 𝑉 ∈ V
9	8	elpw2 5310	. . . . . . . 8 ⊢ (𝐷 ∈ 𝒫 𝑉 ↔ 𝐷 ⊆ 𝑉)
10	9	anbi1i 636	. . . . . . 7 ⊢ ((𝐷 ∈ 𝒫 𝑉 ∧ ◡𝐷 = 𝐷) ↔ (𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷))
11	6, 10	bitri 278	. . . . . 6 ⊢ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ↔ (𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷))
12		elfpw 9321	. . . . . 6 ⊢ (𝐻 ∈ (𝒫 𝐸 ∩ Fin) ↔ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin))
13	11, 12	anbi12i 640	. . . . 5 ⊢ ((𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin)))
14	2, 13	bitri 278	. . . 4 ⊢ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin)))
15	14	anbi1i 636	. . 3 ⊢ ((⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∧ 𝐴 ∈ 𝐸) ↔ (((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin)) ∧ 𝐴 ∈ 𝐸))
16	1, 15	bitri 278	. 2 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ↔ (((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin)) ∧ 𝐴 ∈ 𝐸))
17		df-ot 4603	. . 3 ⊢ ⟨𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴⟩
18		mpstval.e	. . . 4 ⊢ 𝐸 = (mEx‘𝑇)
19		mpstval.p	. . . 4 ⊢ 𝑃 = (mPreSt‘𝑇)
20	7, 18, 19	mpstval 36048	. . 3 ⊢ 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
21	17, 20	eleq12i 2859	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
22		df-3an 1105	. 2 ⊢ (((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ 𝐸) ↔ (((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin)) ∧ 𝐴 ∈ 𝐸))
23	16, 21, 22	3bitr4i 306	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ 𝐸))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {crab 3419   ∩ cin 3907   ⊆ wss 3908  𝒫 cpw 4567  ⟨cop 4600  ⟨cotp 4602   × cxp 5664  ^◡ccnv 5665  ‘cfv 6543  Fincfn 8952  mExcmex 35980  mDVcmdv 35981  mPreStcmpst 35986

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-ot 4603  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-mpst 36006

This theorem is used by:  msrval  36051  msrf  36055  mclsssvlem  36075  mclsax  36082  mclsind  36083  mthmpps  36095  mclsppslem  36096  mclspps  36097