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

Theorem mexval 30653
Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mexval.k 𝐾 = (mTC‘𝑇)
mexval.e 𝐸 = (mEx‘𝑇)
mexval.r 𝑅 = (mREx‘𝑇)
Assertion
Ref Expression
mexval 𝐸 = (𝐾 × 𝑅)

Proof of Theorem mexval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 mexval.e . 2 𝐸 = (mEx‘𝑇)
2 fveq2 6103 . . . . . 6 (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
3 mexval.k . . . . . 6 𝐾 = (mTC‘𝑇)
42, 3syl6eqr 2662 . . . . 5 (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
5 fveq2 6103 . . . . . 6 (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
6 mexval.r . . . . . 6 𝑅 = (mREx‘𝑇)
75, 6syl6eqr 2662 . . . . 5 (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
84, 7xpeq12d 5064 . . . 4 (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅))
9 df-mex 30638 . . . 4 mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
10 fvex 6113 . . . . 5 (mTC‘𝑡) ∈ V
11 fvex 6113 . . . . 5 (mREx‘𝑡) ∈ V
1210, 11xpex 6860 . . . 4 ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V
138, 9, 12fvmpt3i 6196 . . 3 (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
14 xp0 5471 . . . . 5 (𝐾 × ∅) = ∅
1514eqcomi 2619 . . . 4 ∅ = (𝐾 × ∅)
16 fvprc 6097 . . . 4 𝑇 ∈ V → (mEx‘𝑇) = ∅)
17 fvprc 6097 . . . . . 6 𝑇 ∈ V → (mREx‘𝑇) = ∅)
186, 17syl5eq 2656 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
1918xpeq2d 5063 . . . 4 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅))
2015, 16, 193eqtr4a 2670 . . 3 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
2113, 20pm2.61i 175 . 2 (mEx‘𝑇) = (𝐾 × 𝑅)
221, 21eqtri 2632 1 𝐸 = (𝐾 × 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1475  wcel 1977  Vcvv 3173  c0 3874   × cxp 5036  cfv 5804  mTCcmtc 30615  mRExcmrex 30617  mExcmex 30618
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-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-iota 5768  df-fun 5806  df-fv 5812  df-mex 30638
This theorem is referenced by:  mexval2  30654  msubff  30681  msubco  30682  msubff1  30707  mvhf  30709  msubvrs  30711
  Copyright terms: Public domain W3C validator