Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elmpst Structured version   Visualization version   GIF version

Theorem elmpst 34194
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 5673 . . 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 5673 . . . . 5 (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) ↔ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)))
3 cnveq 5833 . . . . . . . . 9 (𝑑 = 𝐷 β†’ ◑𝑑 = ◑𝐷)
4 id 22 . . . . . . . . 9 (𝑑 = 𝐷 β†’ 𝑑 = 𝐷)
53, 4eqeq12d 2749 . . . . . . . 8 (𝑑 = 𝐷 β†’ (◑𝑑 = 𝑑 ↔ ◑𝐷 = 𝐷))
65elrab 3649 . . . . . . 7 (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} ↔ (𝐷 ∈ 𝒫 𝑉 ∧ ◑𝐷 = 𝐷))
7 mpstval.v . . . . . . . . . 10 𝑉 = (mDVβ€˜π‘‡)
87fvexi 6860 . . . . . . . . 9 𝑉 ∈ V
98elpw2 5306 . . . . . . . 8 (𝐷 ∈ 𝒫 𝑉 ↔ 𝐷 βŠ† 𝑉)
109anbi1i 625 . . . . . . 7 ((𝐷 ∈ 𝒫 𝑉 ∧ ◑𝐷 = 𝐷) ↔ (𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷))
116, 10bitri 275 . . . . . 6 (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} ↔ (𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷))
12 elfpw 9304 . . . . . 6 (𝐻 ∈ (𝒫 𝐸 ∩ Fin) ↔ (𝐻 βŠ† 𝐸 ∧ 𝐻 ∈ Fin))
1311, 12anbi12i 628 . . . . 5 ((𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷) ∧ (𝐻 βŠ† 𝐸 ∧ 𝐻 ∈ Fin)))
142, 13bitri 275 . . . 4 (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷 βŠ† 𝑉 ∧ ◑𝐷 = 𝐷) ∧ (𝐻 βŠ† 𝐸 ∧ 𝐻 ∈ Fin)))
1514anbi1i 625 . . 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 275 . 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 4599 . . 3 ⟨𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴⟩
18 mpstval.e . . . 4 𝐸 = (mExβ€˜π‘‡)
19 mpstval.p . . . 4 𝑃 = (mPreStβ€˜π‘‡)
207, 18, 19mpstval 34193 . . 3 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸)
2117, 20eleq12i 2827 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉 ∣ ◑𝑑 = 𝑑} Γ— (𝒫 𝐸 ∩ Fin)) Γ— 𝐸))
22 df-3an 1090 . 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 303 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {crab 3406   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  βŸ¨cop 4596  βŸ¨cotp 4598   Γ— cxp 5635  β—‘ccnv 5636  β€˜cfv 6500  Fincfn 8889  mExcmex 34125  mDVcmdv 34126  mPreStcmpst 34131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-ot 4599  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-mpst 34151
This theorem is referenced by:  msrval  34196  msrf  34200  mclsssvlem  34220  mclsax  34227  mclsind  34228  mthmpps  34240  mclsppslem  34241  mclspps  34242
  Copyright terms: Public domain W3C validator