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Theorem elmthm 29125
Description: A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mthmval.r  |-  R  =  (mStRed `  T )
mthmval.j  |-  J  =  (mPPSt `  T )
mthmval.u  |-  U  =  (mThm `  T )
Assertion
Ref Expression
elmthm  |-  ( X  e.  U  <->  E. x  e.  J  ( R `  x )  =  ( R `  X ) )
Distinct variable groups:    x, J    x, R    x, T    x, X
Allowed substitution hint:    U( x)

Proof of Theorem elmthm
StepHypRef Expression
1 mthmval.r . . . 4  |-  R  =  (mStRed `  T )
2 mthmval.j . . . 4  |-  J  =  (mPPSt `  T )
3 mthmval.u . . . 4  |-  U  =  (mThm `  T )
41, 2, 3mthmval 29124 . . 3  |-  U  =  ( `' R "
( R " J
) )
54eleq2i 2460 . 2  |-  ( X  e.  U  <->  X  e.  ( `' R " ( R
" J ) ) )
6 eqid 2382 . . . . 5  |-  (mPreSt `  T )  =  (mPreSt `  T )
76, 1msrf 29091 . . . 4  |-  R :
(mPreSt `  T ) --> (mPreSt `  T )
8 ffn 5639 . . . 4  |-  ( R : (mPreSt `  T
) --> (mPreSt `  T )  ->  R  Fn  (mPreSt `  T ) )
97, 8ax-mp 5 . . 3  |-  R  Fn  (mPreSt `  T )
10 elpreima 5909 . . 3  |-  ( R  Fn  (mPreSt `  T
)  ->  ( X  e.  ( `' R "
( R " J
) )  <->  ( X  e.  (mPreSt `  T )  /\  ( R `  X
)  e.  ( R
" J ) ) ) )
119, 10ax-mp 5 . 2  |-  ( X  e.  ( `' R " ( R " J
) )  <->  ( X  e.  (mPreSt `  T )  /\  ( R `  X
)  e.  ( R
" J ) ) )
126, 2mppspst 29123 . . . . 5  |-  J  C_  (mPreSt `  T )
13 fvelimab 5830 . . . . 5  |-  ( ( R  Fn  (mPreSt `  T )  /\  J  C_  (mPreSt `  T )
)  ->  ( ( R `  X )  e.  ( R " J
)  <->  E. x  e.  J  ( R `  x )  =  ( R `  X ) ) )
149, 12, 13mp2an 670 . . . 4  |-  ( ( R `  X )  e.  ( R " J )  <->  E. x  e.  J  ( R `  x )  =  ( R `  X ) )
1514anbi2i 692 . . 3  |-  ( ( X  e.  (mPreSt `  T )  /\  ( R `  X )  e.  ( R " J
) )  <->  ( X  e.  (mPreSt `  T )  /\  E. x  e.  J  ( R `  x )  =  ( R `  X ) ) )
1612sseli 3413 . . . . . 6  |-  ( x  e.  J  ->  x  e.  (mPreSt `  T )
)
176, 1msrrcl 29092 . . . . . 6  |-  ( ( R `  x )  =  ( R `  X )  ->  (
x  e.  (mPreSt `  T )  <->  X  e.  (mPreSt `  T ) ) )
1816, 17syl5ibcom 220 . . . . 5  |-  ( x  e.  J  ->  (
( R `  x
)  =  ( R `
 X )  ->  X  e.  (mPreSt `  T
) ) )
1918rexlimiv 2868 . . . 4  |-  ( E. x  e.  J  ( R `  x )  =  ( R `  X )  ->  X  e.  (mPreSt `  T )
)
2019pm4.71ri 631 . . 3  |-  ( E. x  e.  J  ( R `  x )  =  ( R `  X )  <->  ( X  e.  (mPreSt `  T )  /\  E. x  e.  J  ( R `  x )  =  ( R `  X ) ) )
2115, 20bitr4i 252 . 2  |-  ( ( X  e.  (mPreSt `  T )  /\  ( R `  X )  e.  ( R " J
) )  <->  E. x  e.  J  ( R `  x )  =  ( R `  X ) )
225, 11, 213bitri 271 1  |-  ( X  e.  U  <->  E. x  e.  J  ( R `  x )  =  ( R `  X ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   E.wrex 2733    C_ wss 3389   `'ccnv 4912   "cima 4916    Fn wfn 5491   -->wf 5492   ` cfv 5496  mPreStcmpst 29022  mStRedcmsr 29023  mPPStcmpps 29027  mThmcmthm 29028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-ot 3953  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-1st 6699  df-2nd 6700  df-mpst 29042  df-msr 29043  df-mpps 29047  df-mthm 29048
This theorem is referenced by:  mthmi  29126  mthmpps  29131
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