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Theorem mpst123 30691
 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 30690 . . 3 𝑃 ⊆ ((V × V) × V)
32sseli 3564 . 2 (𝑋𝑃𝑋 ∈ ((V × V) × V))
4 1st2nd2 7096 . . . 4 (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
5 xp1st 7089 . . . . . 6 (𝑋 ∈ ((V × V) × V) → (1st𝑋) ∈ (V × V))
6 1st2nd2 7096 . . . . . 6 ((1st𝑋) ∈ (V × V) → (1st𝑋) = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩)
75, 6syl 17 . . . . 5 (𝑋 ∈ ((V × V) × V) → (1st𝑋) = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩)
87opeq1d 4346 . . . 4 (𝑋 ∈ ((V × V) × V) → ⟨(1st𝑋), (2nd𝑋)⟩ = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩)
94, 8eqtrd 2644 . . 3 (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩)
10 df-ot 4134 . . 3 ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋)), (2nd𝑋)⟩ = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩
119, 10syl6eqr 2662 . 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 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ⟨cotp 4133   × cxp 5036  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058  mPreStcmpst 30624 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060  df-mpst 30644 This theorem is referenced by:  msrf  30693  msrid  30696  mthmpps  30733
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