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Theorem mthmval 34864
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 6890 . . . . . . 7 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = (mStRedβ€˜π‘‡))
3 mthmval.r . . . . . . 7 𝑅 = (mStRedβ€˜π‘‡)
42, 3eqtr4di 2788 . . . . . 6 (𝑑 = 𝑇 β†’ (mStRedβ€˜π‘‘) = 𝑅)
54cnveqd 5874 . . . . 5 (𝑑 = 𝑇 β†’ β—‘(mStRedβ€˜π‘‘) = ◑𝑅)
6 fveq2 6890 . . . . . . 7 (𝑑 = 𝑇 β†’ (mPPStβ€˜π‘‘) = (mPPStβ€˜π‘‡))
7 mthmval.j . . . . . . 7 𝐽 = (mPPStβ€˜π‘‡)
86, 7eqtr4di 2788 . . . . . 6 (𝑑 = 𝑇 β†’ (mPPStβ€˜π‘‘) = 𝐽)
94, 8imaeq12d 6059 . . . . 5 (𝑑 = 𝑇 β†’ ((mStRedβ€˜π‘‘) β€œ (mPPStβ€˜π‘‘)) = (𝑅 β€œ 𝐽))
105, 9imaeq12d 6059 . . . 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 34788 . . . 4 mThm = (𝑑 ∈ V ↦ (β—‘(mStRedβ€˜π‘‘) β€œ ((mStRedβ€˜π‘‘) β€œ (mPPStβ€˜π‘‘))))
12 fvex 6903 . . . . . 6 (mStRedβ€˜π‘‘) ∈ V
1312cnvex 7918 . . . . 5 β—‘(mStRedβ€˜π‘‘) ∈ V
14 imaexg 7908 . . . . 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 7002 . . 3 (𝑇 ∈ V β†’ (mThmβ€˜π‘‡) = (◑𝑅 β€œ (𝑅 β€œ 𝐽)))
17 0ima 6076 . . . . 5 (βˆ… β€œ (𝑅 β€œ 𝐽)) = βˆ…
1817eqcomi 2739 . . . 4 βˆ… = (βˆ… β€œ (𝑅 β€œ 𝐽))
19 fvprc 6882 . . . 4 (Β¬ 𝑇 ∈ V β†’ (mThmβ€˜π‘‡) = βˆ…)
20 fvprc 6882 . . . . . . . 8 (Β¬ 𝑇 ∈ V β†’ (mStRedβ€˜π‘‡) = βˆ…)
213, 20eqtrid 2782 . . . . . . 7 (Β¬ 𝑇 ∈ V β†’ 𝑅 = βˆ…)
2221cnveqd 5874 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ ◑𝑅 = β—‘βˆ…)
23 cnv0 6139 . . . . . 6 β—‘βˆ… = βˆ…
2422, 23eqtrdi 2786 . . . . 5 (Β¬ 𝑇 ∈ V β†’ ◑𝑅 = βˆ…)
2524imaeq1d 6057 . . . 4 (Β¬ 𝑇 ∈ V β†’ (◑𝑅 β€œ (𝑅 β€œ 𝐽)) = (βˆ… β€œ (𝑅 β€œ 𝐽)))
2618, 19, 253eqtr4a 2796 . . 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 2758 1 π‘ˆ = (◑𝑅 β€œ (𝑅 β€œ 𝐽))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1539   ∈ wcel 2104  Vcvv 3472  βˆ…c0 4321  β—‘ccnv 5674   β€œ cima 5678  β€˜cfv 6542  mStRedcmsr 34763  mPPStcmpps 34767  mThmcmthm 34768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fv 6550  df-mthm 34788
This theorem is referenced by:  elmthm  34865  mthmsta  34867  mthmblem  34869
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