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

Theorem mthmval 35580
Description: A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. Unlike the difference between pre-statement and statement, this application of the reduct is not necessarily trivial: there are theorems that are not themselves provable but are provable once enough "dummy variables" are introduced. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mthmval.r 𝑅 = (mStRed‘𝑇)
mthmval.j 𝐽 = (mPPSt‘𝑇)
mthmval.u 𝑈 = (mThm‘𝑇)
Assertion
Ref Expression
mthmval 𝑈 = (𝑅 “ (𝑅𝐽))

Proof of Theorem mthmval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 mthmval.u . 2 𝑈 = (mThm‘𝑇)
2 fveq2 6906 . . . . . . 7 (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3 mthmval.r . . . . . . 7 𝑅 = (mStRed‘𝑇)
42, 3eqtr4di 2795 . . . . . 6 (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
54cnveqd 5886 . . . . 5 (𝑡 = 𝑇(mStRed‘𝑡) = 𝑅)
6 fveq2 6906 . . . . . . 7 (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇))
7 mthmval.j . . . . . . 7 𝐽 = (mPPSt‘𝑇)
86, 7eqtr4di 2795 . . . . . 6 (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽)
94, 8imaeq12d 6079 . . . . 5 (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅𝐽))
105, 9imaeq12d 6079 . . . 4 (𝑡 = 𝑇 → ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (𝑅 “ (𝑅𝐽)))
11 df-mthm 35504 . . . 4 mThm = (𝑡 ∈ V ↦ ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
12 fvex 6919 . . . . . 6 (mStRed‘𝑡) ∈ V
1312cnvex 7947 . . . . 5 (mStRed‘𝑡) ∈ V
14 imaexg 7935 . . . . 5 ((mStRed‘𝑡) ∈ V → ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) ∈ V)
1513, 14ax-mp 5 . . . 4 ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) ∈ V
1610, 11, 15fvmpt3i 7021 . . 3 (𝑇 ∈ V → (mThm‘𝑇) = (𝑅 “ (𝑅𝐽)))
17 0ima 6096 . . . . 5 (∅ “ (𝑅𝐽)) = ∅
1817eqcomi 2746 . . . 4 ∅ = (∅ “ (𝑅𝐽))
19 fvprc 6898 . . . 4 𝑇 ∈ V → (mThm‘𝑇) = ∅)
20 fvprc 6898 . . . . . . . 8 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
213, 20eqtrid 2789 . . . . . . 7 𝑇 ∈ V → 𝑅 = ∅)
2221cnveqd 5886 . . . . . 6 𝑇 ∈ V → 𝑅 = ∅)
23 cnv0 6160 . . . . . 6 ∅ = ∅
2422, 23eqtrdi 2793 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
2524imaeq1d 6077 . . . 4 𝑇 ∈ V → (𝑅 “ (𝑅𝐽)) = (∅ “ (𝑅𝐽)))
2618, 19, 253eqtr4a 2803 . . 3 𝑇 ∈ V → (mThm‘𝑇) = (𝑅 “ (𝑅𝐽)))
2716, 26pm2.61i 182 . 2 (mThm‘𝑇) = (𝑅 “ (𝑅𝐽))
281, 27eqtri 2765 1 𝑈 = (𝑅 “ (𝑅𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  ccnv 5684  cima 5688  cfv 6561  mStRedcmsr 35479  mPPStcmpps 35483  mThmcmthm 35484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-mthm 35504
This theorem is referenced by:  elmthm  35581  mthmsta  35583  mthmblem  35585
  Copyright terms: Public domain W3C validator