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Theorem mpst123 31775
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 31774 . . 3 𝑃 ⊆ ((V × V) × V)
32sseli 3748 . 2 (𝑋𝑃𝑋 ∈ ((V × V) × V))
4 1st2nd2 7358 . . . 4 (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
5 xp1st 7351 . . . . . 6 (𝑋 ∈ ((V × V) × V) → (1st𝑋) ∈ (V × V))
6 1st2nd2 7358 . . . . . 6 ((1st𝑋) ∈ (V × V) → (1st𝑋) = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩)
75, 6syl 17 . . . . 5 (𝑋 ∈ ((V × V) × V) → (1st𝑋) = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩)
87opeq1d 4546 . . . 4 (𝑋 ∈ ((V × V) × V) → ⟨(1st𝑋), (2nd𝑋)⟩ = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩)
94, 8eqtrd 2805 . . 3 (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩)
10 df-ot 4326 . . 3 ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋)), (2nd𝑋)⟩ = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩
119, 10syl6eqr 2823 . 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 1631  wcel 2145  Vcvv 3351  cop 4323  cotp 4325   × cxp 5248  cfv 6030  1st c1st 7317  2nd c2nd 7318  mPreStcmpst 31708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-ot 4326  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fv 6038  df-1st 7319  df-2nd 7320  df-mpst 31728
This theorem is referenced by:  msrf  31777  msrid  31780  mthmpps  31817
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