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Theorem mthmval 34503
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 6887 . . . . . . 7 (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3 mthmval.r . . . . . . 7 𝑅 = (mStRed‘𝑇)
42, 3eqtr4di 2791 . . . . . 6 (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
54cnveqd 5872 . . . . 5 (𝑡 = 𝑇(mStRed‘𝑡) = 𝑅)
6 fveq2 6887 . . . . . . 7 (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇))
7 mthmval.j . . . . . . 7 𝐽 = (mPPSt‘𝑇)
86, 7eqtr4di 2791 . . . . . 6 (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽)
94, 8imaeq12d 6057 . . . . 5 (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅𝐽))
105, 9imaeq12d 6057 . . . 4 (𝑡 = 𝑇 → ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (𝑅 “ (𝑅𝐽)))
11 df-mthm 34427 . . . 4 mThm = (𝑡 ∈ V ↦ ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
12 fvex 6900 . . . . . 6 (mStRed‘𝑡) ∈ V
1312cnvex 7910 . . . . 5 (mStRed‘𝑡) ∈ V
14 imaexg 7900 . . . . 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 6998 . . 3 (𝑇 ∈ V → (mThm‘𝑇) = (𝑅 “ (𝑅𝐽)))
17 0ima 6073 . . . . 5 (∅ “ (𝑅𝐽)) = ∅
1817eqcomi 2742 . . . 4 ∅ = (∅ “ (𝑅𝐽))
19 fvprc 6879 . . . 4 𝑇 ∈ V → (mThm‘𝑇) = ∅)
20 fvprc 6879 . . . . . . . 8 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
213, 20eqtrid 2785 . . . . . . 7 𝑇 ∈ V → 𝑅 = ∅)
2221cnveqd 5872 . . . . . 6 𝑇 ∈ V → 𝑅 = ∅)
23 cnv0 6136 . . . . . 6 ∅ = ∅
2422, 23eqtrdi 2789 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
2524imaeq1d 6055 . . . 4 𝑇 ∈ V → (𝑅 “ (𝑅𝐽)) = (∅ “ (𝑅𝐽)))
2618, 19, 253eqtr4a 2799 . . 3 𝑇 ∈ V → (mThm‘𝑇) = (𝑅 “ (𝑅𝐽)))
2716, 26pm2.61i 182 . 2 (mThm‘𝑇) = (𝑅 “ (𝑅𝐽))
281, 27eqtri 2761 1 𝑈 = (𝑅 “ (𝑅𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3475  c0 4320  ccnv 5673  cima 5677  cfv 6539  mStRedcmsr 34402  mPPStcmpps 34406  mThmcmthm 34407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fv 6547  df-mthm 34427
This theorem is referenced by:  elmthm  34504  mthmsta  34506  mthmblem  34508
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