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Theorem elmpst 31162
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 ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin) ∧ 𝐴𝐸))

Proof of Theorem elmpst
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 opelxp 5108 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ↔ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∧ 𝐴𝐸))
2 opelxp 5108 . . . . 5 (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)))
3 cnveq 5258 . . . . . . . . 9 (𝑑 = 𝐷𝑑 = 𝐷)
4 id 22 . . . . . . . . 9 (𝑑 = 𝐷𝑑 = 𝐷)
53, 4eqeq12d 2636 . . . . . . . 8 (𝑑 = 𝐷 → (𝑑 = 𝑑𝐷 = 𝐷))
65elrab 3347 . . . . . . 7 (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ↔ (𝐷 ∈ 𝒫 𝑉𝐷 = 𝐷))
7 mpstval.v . . . . . . . . . 10 𝑉 = (mDV‘𝑇)
8 fvex 6160 . . . . . . . . . 10 (mDV‘𝑇) ∈ V
97, 8eqeltri 2694 . . . . . . . . 9 𝑉 ∈ V
109elpw2 4790 . . . . . . . 8 (𝐷 ∈ 𝒫 𝑉𝐷𝑉)
1110anbi1i 730 . . . . . . 7 ((𝐷 ∈ 𝒫 𝑉𝐷 = 𝐷) ↔ (𝐷𝑉𝐷 = 𝐷))
126, 11bitri 264 . . . . . 6 (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ↔ (𝐷𝑉𝐷 = 𝐷))
13 elfpw 8215 . . . . . 6 (𝐻 ∈ (𝒫 𝐸 ∩ Fin) ↔ (𝐻𝐸𝐻 ∈ Fin))
1412, 13anbi12i 732 . . . . 5 ((𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)))
152, 14bitri 264 . . . 4 (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)))
1615anbi1i 730 . . 3 ((⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∧ 𝐴𝐸) ↔ (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)) ∧ 𝐴𝐸))
171, 16bitri 264 . 2 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ↔ (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)) ∧ 𝐴𝐸))
18 df-ot 4159 . . 3 𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴
19 mpstval.e . . . 4 𝐸 = (mEx‘𝑇)
20 mpstval.p . . . 4 𝑃 = (mPreSt‘𝑇)
217, 19, 20mpstval 31161 . . 3 𝑃 = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
2218, 21eleq12i 2691 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
23 df-3an 1038 . 2 (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin) ∧ 𝐴𝐸) ↔ (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)) ∧ 𝐴𝐸))
2417, 22, 233bitr4i 292 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin) ∧ 𝐴𝐸))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  cin 3555  wss 3556  𝒫 cpw 4132  cop 4156  cotp 4158   × cxp 5074  ccnv 5075  cfv 5849  Fincfn 7902  mExcmex 31093  mDVcmdv 31094  mPreStcmpst 31099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-ot 4159  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-iota 5812  df-fun 5851  df-fv 5857  df-mpst 31119
This theorem is referenced by:  msrval  31164  msrf  31168  mclsssvlem  31188  mclsax  31195  mclsind  31196  mthmpps  31208  mclsppslem  31209  mclspps  31210
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