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Theorem mthmval 35121
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 = (mStRedβ€˜π‘‡)
mthmval.j 𝐽 = (mPPStβ€˜π‘‡)
mthmval.u π‘ˆ = (mThmβ€˜π‘‡)
Assertion
Ref Expression
mthmval π‘ˆ = (◑𝑅 β€œ (𝑅 β€œ 𝐽))

Proof of Theorem mthmval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 mthmval.u . 2 π‘ˆ = (mThmβ€˜π‘‡)
2 fveq2 6891 . . . . . . 7 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = (mStRedβ€˜π‘‡))
3 mthmval.r . . . . . . 7 𝑅 = (mStRedβ€˜π‘‡)
42, 3eqtr4di 2785 . . . . . 6 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = 𝑅)
54cnveqd 5872 . . . . 5 (𝑑 = 𝑇 β†’ β—‘(mStRedβ€˜π‘‘) = ◑𝑅)
6 fveq2 6891 . . . . . . 7 (𝑑 = 𝑇 β†’ (mPPStβ€˜π‘‘) = (mPPStβ€˜π‘‡))
7 mthmval.j . . . . . . 7 𝐽 = (mPPStβ€˜π‘‡)
86, 7eqtr4di 2785 . . . . . 6 (𝑑 = 𝑇 β†’ (mPPStβ€˜π‘‘) = 𝐽)
94, 8imaeq12d 6058 . . . . 5 (𝑑 = 𝑇 β†’ ((mStRedβ€˜π‘‘) β€œ (mPPStβ€˜π‘‘)) = (𝑅 β€œ 𝐽))
105, 9imaeq12d 6058 . . . 4 (𝑑 = 𝑇 β†’ (β—‘(mStRedβ€˜π‘‘) β€œ ((mStRedβ€˜π‘‘) β€œ (mPPStβ€˜π‘‘))) = (◑𝑅 β€œ (𝑅 β€œ 𝐽)))
11 df-mthm 35045 . . . 4 mThm = (𝑑 ∈ V ↦ (β—‘(mStRedβ€˜π‘‘) β€œ ((mStRedβ€˜π‘‘) β€œ (mPPStβ€˜π‘‘))))
12 fvex 6904 . . . . . 6 (mStRedβ€˜π‘‘) ∈ V
1312cnvex 7927 . . . . 5 β—‘(mStRedβ€˜π‘‘) ∈ V
14 imaexg 7915 . . . . 5 (β—‘(mStRedβ€˜π‘‘) ∈ V β†’ (β—‘(mStRedβ€˜π‘‘) β€œ ((mStRedβ€˜π‘‘) β€œ (mPPStβ€˜π‘‘))) ∈ V)
1513, 14ax-mp 5 . . . 4 (β—‘(mStRedβ€˜π‘‘) β€œ ((mStRedβ€˜π‘‘) β€œ (mPPStβ€˜π‘‘))) ∈ V
1610, 11, 15fvmpt3i 7004 . . 3 (𝑇 ∈ V β†’ (mThmβ€˜π‘‡) = (◑𝑅 β€œ (𝑅 β€œ 𝐽)))
17 0ima 6075 . . . . 5 (βˆ… β€œ (𝑅 β€œ 𝐽)) = βˆ…
1817eqcomi 2736 . . . 4 βˆ… = (βˆ… β€œ (𝑅 β€œ 𝐽))
19 fvprc 6883 . . . 4 (Β¬ 𝑇 ∈ V β†’ (mThmβ€˜π‘‡) = βˆ…)
20 fvprc 6883 . . . . . . . 8 (Β¬ 𝑇 ∈ V β†’ (mStRedβ€˜π‘‡) = βˆ…)
213, 20eqtrid 2779 . . . . . . 7 (Β¬ 𝑇 ∈ V β†’ 𝑅 = βˆ…)
2221cnveqd 5872 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ ◑𝑅 = β—‘βˆ…)
23 cnv0 6139 . . . . . 6 β—‘βˆ… = βˆ…
2422, 23eqtrdi 2783 . . . . 5 (Β¬ 𝑇 ∈ V β†’ ◑𝑅 = βˆ…)
2524imaeq1d 6056 . . . 4 (Β¬ 𝑇 ∈ V β†’ (◑𝑅 β€œ (𝑅 β€œ 𝐽)) = (βˆ… β€œ (𝑅 β€œ 𝐽)))
2618, 19, 253eqtr4a 2793 . . 3 (Β¬ 𝑇 ∈ V β†’ (mThmβ€˜π‘‡) = (◑𝑅 β€œ (𝑅 β€œ 𝐽)))
2716, 26pm2.61i 182 . 2 (mThmβ€˜π‘‡) = (◑𝑅 β€œ (𝑅 β€œ 𝐽))
281, 27eqtri 2755 1 π‘ˆ = (◑𝑅 β€œ (𝑅 β€œ 𝐽))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1534   ∈ wcel 2099  Vcvv 3469  βˆ…c0 4318  β—‘ccnv 5671   β€œ cima 5675  β€˜cfv 6542  mStRedcmsr 35020  mPPStcmpps 35024  mThmcmthm 35025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-mthm 35045
This theorem is referenced by:  elmthm  35122  mthmsta  35124  mthmblem  35126
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