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Theorem mppsthm 31181
Description: A provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mppsthm.j 𝐽 = (mPPSt‘𝑇)
mppsthm.u 𝑈 = (mThm‘𝑇)
Assertion
Ref Expression
mppsthm 𝐽𝑈

Proof of Theorem mppsthm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . 3 ((mStRed‘𝑇)‘𝑥) = ((mStRed‘𝑇)‘𝑥)
2 eqid 2621 . . . 4 (mStRed‘𝑇) = (mStRed‘𝑇)
3 mppsthm.j . . . 4 𝐽 = (mPPSt‘𝑇)
4 mppsthm.u . . . 4 𝑈 = (mThm‘𝑇)
52, 3, 4mthmi 31179 . . 3 ((𝑥𝐽 ∧ ((mStRed‘𝑇)‘𝑥) = ((mStRed‘𝑇)‘𝑥)) → 𝑥𝑈)
61, 5mpan2 706 . 2 (𝑥𝐽𝑥𝑈)
76ssriv 3587 1 𝐽𝑈
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  wss 3555  cfv 5847  mStRedcmsr 31076  mPPStcmpps 31080  mThmcmthm 31081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-ot 4157  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-1st 7113  df-2nd 7114  df-mpst 31095  df-msr 31096  df-mpps 31100  df-mthm 31101
This theorem is referenced by: (None)
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