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Theorem mpst123 35281
Description: Decompose a pre-statement into a triple of values. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
mpstssv.p 𝑃 = (mPreSt‘𝑇)
Assertion
Ref Expression
mpst123 (𝑋𝑃𝑋 = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋)), (2nd𝑋)⟩)

Proof of Theorem mpst123
StepHypRef Expression
1 mpstssv.p . . . 4 𝑃 = (mPreSt‘𝑇)
21mpstssv 35280 . . 3 𝑃 ⊆ ((V × V) × V)
32sseli 3972 . 2 (𝑋𝑃𝑋 ∈ ((V × V) × V))
4 1st2nd2 8033 . . . 4 (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
5 xp1st 8026 . . . . . 6 (𝑋 ∈ ((V × V) × V) → (1st𝑋) ∈ (V × V))
6 1st2nd2 8033 . . . . . 6 ((1st𝑋) ∈ (V × V) → (1st𝑋) = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩)
75, 6syl 17 . . . . 5 (𝑋 ∈ ((V × V) × V) → (1st𝑋) = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩)
87opeq1d 4881 . . . 4 (𝑋 ∈ ((V × V) × V) → ⟨(1st𝑋), (2nd𝑋)⟩ = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩)
94, 8eqtrd 2765 . . 3 (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩)
10 df-ot 4639 . . 3 ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋)), (2nd𝑋)⟩ = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩
119, 10eqtr4di 2783 . 2 (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋)), (2nd𝑋)⟩)
123, 11syl 17 1 (𝑋𝑃𝑋 = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋)), (2nd𝑋)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3461  cop 4636  cotp 4638   × cxp 5676  cfv 6549  1st c1st 7992  2nd c2nd 7993  mPreStcmpst 35214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-ot 4639  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fv 6557  df-1st 7994  df-2nd 7995  df-mpst 35234
This theorem is referenced by:  msrf  35283  msrid  35286  mthmpps  35323
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