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Theorem elmpst 34522
Description: Property of being a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mpstval.v 𝑉 = (mDVβ€˜π‘‡)
mpstval.e 𝐸 = (mExβ€˜π‘‡)
mpstval.p 𝑃 = (mPreStβ€˜π‘‡)
Assertion
Ref Expression
elmpst (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷) ∧ (𝐻 βŠ† 𝐸 ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ 𝐸))

Proof of Theorem elmpst
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 opelxp 5712 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸) ↔ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) ∧ 𝐴 ∈ 𝐸))
2 opelxp 5712 . . . . 5 (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) ↔ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)))
3 cnveq 5873 . . . . . . . . 9 (𝑑 = 𝐷 β†’ ◑𝑑 = ◑𝐷)
4 id 22 . . . . . . . . 9 (𝑑 = 𝐷 β†’ 𝑑 = 𝐷)
53, 4eqeq12d 2748 . . . . . . . 8 (𝑑 = 𝐷 β†’ (◑𝑑 = 𝑑 ↔ ◑𝐷 = 𝐷))
65elrab 3683 . . . . . . 7 (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} ↔ (𝐷 ∈ 𝒫 𝑉 ∧ ◑𝐷 = 𝐷))
7 mpstval.v . . . . . . . . . 10 𝑉 = (mDVβ€˜π‘‡)
87fvexi 6905 . . . . . . . . 9 𝑉 ∈ V
98elpw2 5345 . . . . . . . 8 (𝐷 ∈ 𝒫 𝑉 ↔ 𝐷 βŠ† 𝑉)
109anbi1i 624 . . . . . . 7 ((𝐷 ∈ 𝒫 𝑉 ∧ ◑𝐷 = 𝐷) ↔ (𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷))
116, 10bitri 274 . . . . . 6 (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} ↔ (𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷))
12 elfpw 9353 . . . . . 6 (𝐻 ∈ (𝒫 𝐸 ∩ Fin) ↔ (𝐻 βŠ† 𝐸 ∧ 𝐻 ∈ Fin))
1311, 12anbi12i 627 . . . . 5 ((𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷) ∧ (𝐻 βŠ† 𝐸 ∧ 𝐻 ∈ Fin)))
142, 13bitri 274 . . . 4 (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷) ∧ (𝐻 βŠ† 𝐸 ∧ 𝐻 ∈ Fin)))
1514anbi1i 624 . . 3 ((⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) ∧ 𝐴 ∈ 𝐸) ↔ (((𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷) ∧ (𝐻 βŠ† 𝐸 ∧ 𝐻 ∈ Fin)) ∧ 𝐴 ∈ 𝐸))
161, 15bitri 274 . 2 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸) ↔ (((𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷) ∧ (𝐻 βŠ† 𝐸 ∧ 𝐻 ∈ Fin)) ∧ 𝐴 ∈ 𝐸))
17 df-ot 4637 . . 3 ⟨𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴⟩
18 mpstval.e . . . 4 𝐸 = (mExβ€˜π‘‡)
19 mpstval.p . . . 4 𝑃 = (mPreStβ€˜π‘‡)
207, 18, 19mpstval 34521 . . 3 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸)
2117, 20eleq12i 2826 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸))
22 df-3an 1089 . 2 (((𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷) ∧ (𝐻 βŠ† 𝐸 ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ 𝐸) ↔ (((𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷) ∧ (𝐻 βŠ† 𝐸 ∧ 𝐻 ∈ Fin)) ∧ 𝐴 ∈ 𝐸))
2316, 21, 223bitr4i 302 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷) ∧ (𝐻 βŠ† 𝐸 ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ 𝐸))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3432   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  βŸ¨cop 4634  βŸ¨cotp 4636   Γ— cxp 5674  β—‘ccnv 5675  β€˜cfv 6543  Fincfn 8938  mExcmex 34453  mDVcmdv 34454  mPreStcmpst 34459
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-mpst 34479
This theorem is referenced by:  msrval  34524  msrf  34528  mclsssvlem  34548  mclsax  34555  mclsind  34556  mthmpps  34568  mclsppslem  34569  mclspps  34570
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