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Theorem mthmval 30726
 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 6103 . . . . . . 7 (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3 mthmval.r . . . . . . 7 𝑅 = (mStRed‘𝑇)
42, 3syl6eqr 2662 . . . . . 6 (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
54cnveqd 5220 . . . . 5 (𝑡 = 𝑇(mStRed‘𝑡) = 𝑅)
6 fveq2 6103 . . . . . . 7 (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇))
7 mthmval.j . . . . . . 7 𝐽 = (mPPSt‘𝑇)
86, 7syl6eqr 2662 . . . . . 6 (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽)
94, 8imaeq12d 5386 . . . . 5 (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅𝐽))
105, 9imaeq12d 5386 . . . 4 (𝑡 = 𝑇 → ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (𝑅 “ (𝑅𝐽)))
11 df-mthm 30650 . . . 4 mThm = (𝑡 ∈ V ↦ ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
12 fvex 6113 . . . . . 6 (mStRed‘𝑡) ∈ V
1312cnvex 7006 . . . . 5 (mStRed‘𝑡) ∈ V
14 imaexg 6995 . . . . 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 6196 . . 3 (𝑇 ∈ V → (mThm‘𝑇) = (𝑅 “ (𝑅𝐽)))
17 0ima 5401 . . . . 5 (∅ “ (𝑅𝐽)) = ∅
1817eqcomi 2619 . . . 4 ∅ = (∅ “ (𝑅𝐽))
19 fvprc 6097 . . . 4 𝑇 ∈ V → (mThm‘𝑇) = ∅)
20 fvprc 6097 . . . . . . . 8 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
213, 20syl5eq 2656 . . . . . . 7 𝑇 ∈ V → 𝑅 = ∅)
2221cnveqd 5220 . . . . . 6 𝑇 ∈ V → 𝑅 = ∅)
23 cnv0 5454 . . . . . 6 ∅ = ∅
2422, 23syl6eq 2660 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
2524imaeq1d 5384 . . . 4 𝑇 ∈ V → (𝑅 “ (𝑅𝐽)) = (∅ “ (𝑅𝐽)))
2618, 19, 253eqtr4a 2670 . . 3 𝑇 ∈ V → (mThm‘𝑇) = (𝑅 “ (𝑅𝐽)))
2716, 26pm2.61i 175 . 2 (mThm‘𝑇) = (𝑅 “ (𝑅𝐽))
281, 27eqtri 2632 1 𝑈 = (𝑅 “ (𝑅𝐽))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ◡ccnv 5037   “ cima 5041  ‘cfv 5804  mStRedcmsr 30625  mPPStcmpps 30629  mThmcmthm 30630 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-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-mthm 30650 This theorem is referenced by:  elmthm  30727  mthmsta  30729  mthmblem  30731
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