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Theorem mthmsta 29205
Description: A theorem is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mthmsta.u  |-  U  =  (mThm `  T )
mthmsta.s  |-  S  =  (mPreSt `  T )
Assertion
Ref Expression
mthmsta  |-  U  C_  S

Proof of Theorem mthmsta
StepHypRef Expression
1 eqid 2454 . . 3  |-  (mStRed `  T )  =  (mStRed `  T )
2 eqid 2454 . . 3  |-  (mPPSt `  T )  =  (mPPSt `  T )
3 mthmsta.u . . 3  |-  U  =  (mThm `  T )
41, 2, 3mthmval 29202 . 2  |-  U  =  ( `' (mStRed `  T ) " (
(mStRed `  T ) " (mPPSt `  T )
) )
5 cnvimass 5345 . . 3  |-  ( `' (mStRed `  T ) " ( (mStRed `  T ) " (mPPSt `  T ) ) ) 
C_  dom  (mStRed `  T
)
6 mthmsta.s . . . . 5  |-  S  =  (mPreSt `  T )
76, 1msrf 29169 . . . 4  |-  (mStRed `  T ) : S --> S
87fdmi 5718 . . 3  |-  dom  (mStRed `  T )  =  S
95, 8sseqtri 3521 . 2  |-  ( `' (mStRed `  T ) " ( (mStRed `  T ) " (mPPSt `  T ) ) ) 
C_  S
104, 9eqsstri 3519 1  |-  U  C_  S
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    C_ wss 3461   `'ccnv 4987   dom cdm 4988   "cima 4991   ` cfv 5570  mPreStcmpst 29100  mStRedcmsr 29101  mPPStcmpps 29105  mThmcmthm 29106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-ot 4025  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-1st 6773  df-2nd 6774  df-mpst 29120  df-msr 29121  df-mthm 29126
This theorem is referenced by:  mthmpps  29209
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