Theorem elmpst 35734

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 5660	. . 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 5660	. . . . 5 ⊢ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)))
3		cnveq 5822	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ◡𝑑 = ◡𝐷)
4		id 22	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → 𝑑 = 𝐷)
5	3, 4	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (◡𝑑 = 𝑑 ↔ ◡𝐷 = 𝐷))
6	5	elrab 3635	. . . . . . 7 ⊢ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ↔ (𝐷 ∈ 𝒫 𝑉 ∧ ◡𝐷 = 𝐷))
7		mpstval.v	. . . . . . . . . 10 ⊢ 𝑉 = (mDV‘𝑇)
8	7	fvexi 6848	. . . . . . . . 9 ⊢ 𝑉 ∈ V
9	8	elpw2 5271	. . . . . . . 8 ⊢ (𝐷 ∈ 𝒫 𝑉 ↔ 𝐷 ⊆ 𝑉)
10	9	anbi1i 625	. . . . . . 7 ⊢ ((𝐷 ∈ 𝒫 𝑉 ∧ ◡𝐷 = 𝐷) ↔ (𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷))
11	6, 10	bitri 275	. . . . . 6 ⊢ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ↔ (𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷))
12		elfpw 9257	. . . . . 6 ⊢ (𝐻 ∈ (𝒫 𝐸 ∩ Fin) ↔ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin))
13	11, 12	anbi12i 629	. . . . 5 ⊢ ((𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin)))
14	2, 13	bitri 275	. . . 4 ⊢ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin)))
15	14	anbi1i 625	. . 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 275	. 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 4577	. . 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 35733	. . 3 ⊢ 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
21	17, 20	eleq12i 2830	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
22		df-3an 1089	. 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 303	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ 𝐸))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {crab 3390   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ⟨cop 4574  ⟨cotp 4576   × cxp 5622  ^◡ccnv 5623  ‘cfv 6492  Fincfn 8886  mExcmex 35665  mDVcmdv 35666  mPreStcmpst 35671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-ot 4577  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-mpst 35691

This theorem is referenced by:  msrval  35736  msrf  35740  mclsssvlem  35760  mclsax  35767  mclsind  35768  mthmpps  35780  mclsppslem  35781  mclspps  35782