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Theorem mthmval 35272
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 6894 . . . . . . 7 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = (mStRedβ€˜π‘‡))
3 mthmval.r . . . . . . 7 𝑅 = (mStRedβ€˜π‘‡)
42, 3eqtr4di 2783 . . . . . 6 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = 𝑅)
54cnveqd 5877 . . . . 5 (𝑑 = 𝑇 β†’ β—‘(mStRedβ€˜π‘‘) = ◑𝑅)
6 fveq2 6894 . . . . . . 7 (𝑑 = 𝑇 β†’ (mPPStβ€˜π‘‘) = (mPPStβ€˜π‘‡))
7 mthmval.j . . . . . . 7 𝐽 = (mPPStβ€˜π‘‡)
86, 7eqtr4di 2783 . . . . . 6 (𝑑 = 𝑇 β†’ (mPPStβ€˜π‘‘) = 𝐽)
94, 8imaeq12d 6064 . . . . 5 (𝑑 = 𝑇 β†’ ((mStRedβ€˜π‘‘) β€œ (mPPStβ€˜π‘‘)) = (𝑅 β€œ 𝐽))
105, 9imaeq12d 6064 . . . 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 35196 . . . 4 mThm = (𝑑 ∈ V ↦ (β—‘(mStRedβ€˜π‘‘) β€œ ((mStRedβ€˜π‘‘) β€œ (mPPStβ€˜π‘‘))))
12 fvex 6907 . . . . . 6 (mStRedβ€˜π‘‘) ∈ V
1312cnvex 7931 . . . . 5 β—‘(mStRedβ€˜π‘‘) ∈ V
14 imaexg 7919 . . . . 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 7007 . . 3 (𝑇 ∈ V β†’ (mThmβ€˜π‘‡) = (◑𝑅 β€œ (𝑅 β€œ 𝐽)))
17 0ima 6081 . . . . 5 (βˆ… β€œ (𝑅 β€œ 𝐽)) = βˆ…
1817eqcomi 2734 . . . 4 βˆ… = (βˆ… β€œ (𝑅 β€œ 𝐽))
19 fvprc 6886 . . . 4 (Β¬ 𝑇 ∈ V β†’ (mThmβ€˜π‘‡) = βˆ…)
20 fvprc 6886 . . . . . . . 8 (Β¬ 𝑇 ∈ V β†’ (mStRedβ€˜π‘‡) = βˆ…)
213, 20eqtrid 2777 . . . . . . 7 (Β¬ 𝑇 ∈ V β†’ 𝑅 = βˆ…)
2221cnveqd 5877 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ ◑𝑅 = β—‘βˆ…)
23 cnv0 6145 . . . . . 6 β—‘βˆ… = βˆ…
2422, 23eqtrdi 2781 . . . . 5 (Β¬ 𝑇 ∈ V β†’ ◑𝑅 = βˆ…)
2524imaeq1d 6062 . . . 4 (Β¬ 𝑇 ∈ V β†’ (◑𝑅 β€œ (𝑅 β€œ 𝐽)) = (βˆ… β€œ (𝑅 β€œ 𝐽)))
2618, 19, 253eqtr4a 2791 . . 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 2753 1 π‘ˆ = (◑𝑅 β€œ (𝑅 β€œ 𝐽))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1533   ∈ wcel 2098  Vcvv 3463  βˆ…c0 4323  β—‘ccnv 5676   β€œ cima 5680  β€˜cfv 6547  mStRedcmsr 35171  mPPStcmpps 35175  mThmcmthm 35176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6499  df-fun 6549  df-fv 6555  df-mthm 35196
This theorem is referenced by:  elmthm  35273  mthmsta  35275  mthmblem  35277
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