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Theorem mppsthm 33254
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 2737 . . 3 ((mStRed‘𝑇)‘𝑥) = ((mStRed‘𝑇)‘𝑥)
2 eqid 2737 . . . 4 (mStRed‘𝑇) = (mStRed‘𝑇)
3 mppsthm.j . . . 4 𝐽 = (mPPSt‘𝑇)
4 mppsthm.u . . . 4 𝑈 = (mThm‘𝑇)
52, 3, 4mthmi 33252 . . 3 ((𝑥𝐽 ∧ ((mStRed‘𝑇)‘𝑥) = ((mStRed‘𝑇)‘𝑥)) → 𝑥𝑈)
61, 5mpan2 691 . 2 (𝑥𝐽𝑥𝑈)
76ssriv 3905 1 𝐽𝑈
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wcel 2110  wss 3866  cfv 6380  mStRedcmsr 33149  mPPStcmpps 33153  mThmcmthm 33154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-ot 4550  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-1st 7761  df-2nd 7762  df-mpst 33168  df-msr 33169  df-mpps 33173  df-mthm 33174
This theorem is referenced by: (None)
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