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Theorem mthmval 31800
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 6411 . . . . . . 7 (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3 mthmval.r . . . . . . 7 𝑅 = (mStRed‘𝑇)
42, 3syl6eqr 2865 . . . . . 6 (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
54cnveqd 5506 . . . . 5 (𝑡 = 𝑇(mStRed‘𝑡) = 𝑅)
6 fveq2 6411 . . . . . . 7 (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇))
7 mthmval.j . . . . . . 7 𝐽 = (mPPSt‘𝑇)
86, 7syl6eqr 2865 . . . . . 6 (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽)
94, 8imaeq12d 5684 . . . . 5 (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅𝐽))
105, 9imaeq12d 5684 . . . 4 (𝑡 = 𝑇 → ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (𝑅 “ (𝑅𝐽)))
11 df-mthm 31724 . . . 4 mThm = (𝑡 ∈ V ↦ ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
12 fvex 6424 . . . . . 6 (mStRed‘𝑡) ∈ V
1312cnvex 7346 . . . . 5 (mStRed‘𝑡) ∈ V
14 imaexg 7336 . . . . 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 6511 . . 3 (𝑇 ∈ V → (mThm‘𝑇) = (𝑅 “ (𝑅𝐽)))
17 0ima 5699 . . . . 5 (∅ “ (𝑅𝐽)) = ∅
1817eqcomi 2822 . . . 4 ∅ = (∅ “ (𝑅𝐽))
19 fvprc 6404 . . . 4 𝑇 ∈ V → (mThm‘𝑇) = ∅)
20 fvprc 6404 . . . . . . . 8 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
213, 20syl5eq 2859 . . . . . . 7 𝑇 ∈ V → 𝑅 = ∅)
2221cnveqd 5506 . . . . . 6 𝑇 ∈ V → 𝑅 = ∅)
23 cnv0 5753 . . . . . 6 ∅ = ∅
2422, 23syl6eq 2863 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
2524imaeq1d 5682 . . . 4 𝑇 ∈ V → (𝑅 “ (𝑅𝐽)) = (∅ “ (𝑅𝐽)))
2618, 19, 253eqtr4a 2873 . . 3 𝑇 ∈ V → (mThm‘𝑇) = (𝑅 “ (𝑅𝐽)))
2716, 26pm2.61i 176 . 2 (mThm‘𝑇) = (𝑅 “ (𝑅𝐽))
281, 27eqtri 2835 1 𝑈 = (𝑅 “ (𝑅𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1637  wcel 2157  Vcvv 3398  c0 4123  ccnv 5317  cima 5321  cfv 6104  mStRedcmsr 31699  mPPStcmpps 31703  mThmcmthm 31704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-iota 6067  df-fun 6106  df-fv 6112  df-mthm 31724
This theorem is referenced by:  elmthm  31801  mthmsta  31803  mthmblem  31805
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