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Theorem pmtrdifel 16376
Description: A transposition of elements of a set without a special element corresponds to a transposition of elements of the set. (Contributed by AV, 15-Jan-2019.)
Hypotheses
Ref Expression
pmtrdifel.t  |-  T  =  ran  (pmTrsp `  ( N  \  { K }
) )
pmtrdifel.r  |-  R  =  ran  (pmTrsp `  N
)
Assertion
Ref Expression
pmtrdifel  |-  A. t  e.  T  E. r  e.  R  A. x  e.  ( N  \  { K } ) ( t `
 x )  =  ( r `  x
)
Distinct variable groups:    t, r, x    K, r    N, r, x    R, r    x, T
Allowed substitution hints:    R( x, t)    T( t, r)    K( x, t)    N( t)

Proof of Theorem pmtrdifel
StepHypRef Expression
1 pmtrdifel.t . . . 4  |-  T  =  ran  (pmTrsp `  ( N  \  { K }
) )
2 pmtrdifel.r . . . 4  |-  R  =  ran  (pmTrsp `  N
)
3 eqid 2467 . . . 4  |-  ( (pmTrsp `  N ) `  dom  ( t  \  _I  ) )  =  ( (pmTrsp `  N ) `  dom  ( t  \  _I  ) )
41, 2, 3pmtrdifellem1 16372 . . 3  |-  ( t  e.  T  ->  (
(pmTrsp `  N ) `  dom  ( t  \  _I  ) )  e.  R
)
51, 2, 3pmtrdifellem3 16374 . . 3  |-  ( t  e.  T  ->  A. x  e.  ( N  \  { K } ) ( t `
 x )  =  ( ( (pmTrsp `  N ) `  dom  ( t  \  _I  ) ) `  x
) )
6 fveq1 5871 . . . . . 6  |-  ( r  =  ( (pmTrsp `  N ) `  dom  ( t  \  _I  ) )  ->  (
r `  x )  =  ( ( (pmTrsp `  N ) `  dom  ( t  \  _I  ) ) `  x
) )
76eqeq2d 2481 . . . . 5  |-  ( r  =  ( (pmTrsp `  N ) `  dom  ( t  \  _I  ) )  ->  (
( t `  x
)  =  ( r `
 x )  <->  ( t `  x )  =  ( ( (pmTrsp `  N
) `  dom  ( t 
\  _I  ) ) `
 x ) ) )
87ralbidv 2906 . . . 4  |-  ( r  =  ( (pmTrsp `  N ) `  dom  ( t  \  _I  ) )  ->  ( A. x  e.  ( N  \  { K }
) ( t `  x )  =  ( r `  x )  <->  A. x  e.  ( N  \  { K }
) ( t `  x )  =  ( ( (pmTrsp `  N
) `  dom  ( t 
\  _I  ) ) `
 x ) ) )
98rspcev 3219 . . 3  |-  ( ( ( (pmTrsp `  N
) `  dom  ( t 
\  _I  ) )  e.  R  /\  A. x  e.  ( N  \  { K } ) ( t `  x
)  =  ( ( (pmTrsp `  N ) `  dom  ( t  \  _I  ) ) `  x
) )  ->  E. r  e.  R  A. x  e.  ( N  \  { K } ) ( t `
 x )  =  ( r `  x
) )
104, 5, 9syl2anc 661 . 2  |-  ( t  e.  T  ->  E. r  e.  R  A. x  e.  ( N  \  { K } ) ( t `
 x )  =  ( r `  x
) )
1110rgen 2827 1  |-  A. t  e.  T  E. r  e.  R  A. x  e.  ( N  \  { K } ) ( t `
 x )  =  ( r `  x
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    \ cdif 3478   {csn 4033    _I cid 4796   dom cdm 5005   ran crn 5006   ` cfv 5594  pmTrspcpmtr 16337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-1o 7142  df-2o 7143  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pmtr 16338
This theorem is referenced by: (None)
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