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Theorem pmtrdifellem2 16101
Description: Lemma 2 for pmtrdifel 16104. (Contributed by AV, 15-Jan-2019.)
Hypotheses
Ref Expression
pmtrdifel.t  |-  T  =  ran  (pmTrsp `  ( N  \  { K }
) )
pmtrdifel.r  |-  R  =  ran  (pmTrsp `  N
)
pmtrdifel.0  |-  S  =  ( (pmTrsp `  N
) `  dom  ( Q 
\  _I  ) )
Assertion
Ref Expression
pmtrdifellem2  |-  ( Q  e.  T  ->  dom  ( S  \  _I  )  =  dom  ( Q  \  _I  ) )

Proof of Theorem pmtrdifellem2
StepHypRef Expression
1 pmtrdifel.0 . . . 4  |-  S  =  ( (pmTrsp `  N
) `  dom  ( Q 
\  _I  ) )
21difeq1i 3577 . . 3  |-  ( S 
\  _I  )  =  ( ( (pmTrsp `  N ) `  dom  ( Q  \  _I  )
)  \  _I  )
32dmeqi 5148 . 2  |-  dom  ( S  \  _I  )  =  dom  ( ( (pmTrsp `  N ) `  dom  ( Q  \  _I  )
)  \  _I  )
4 eqid 2454 . . . . 5  |-  (pmTrsp `  ( N  \  { K } ) )  =  (pmTrsp `  ( N  \  { K } ) )
5 pmtrdifel.t . . . . 5  |-  T  =  ran  (pmTrsp `  ( N  \  { K }
) )
64, 5pmtrfb 16089 . . . 4  |-  ( Q  e.  T  <->  ( ( N  \  { K }
)  e.  _V  /\  Q : ( N  \  { K } ) -1-1-onto-> ( N 
\  { K }
)  /\  dom  ( Q 
\  _I  )  ~~  2o ) )
7 difsnexi 6493 . . . . 5  |-  ( ( N  \  { K } )  e.  _V  ->  N  e.  _V )
8 f1of 5748 . . . . . 6  |-  ( Q : ( N  \  { K } ) -1-1-onto-> ( N 
\  { K }
)  ->  Q :
( N  \  { K } ) --> ( N 
\  { K }
) )
9 fdm 5670 . . . . . 6  |-  ( Q : ( N  \  { K } ) --> ( N  \  { K } )  ->  dom  Q  =  ( N  \  { K } ) )
10 difssd 3591 . . . . . . . 8  |-  ( dom 
Q  =  ( N 
\  { K }
)  ->  ( Q  \  _I  )  C_  Q
)
11 dmss 5146 . . . . . . . 8  |-  ( ( Q  \  _I  )  C_  Q  ->  dom  ( Q 
\  _I  )  C_  dom  Q )
1210, 11syl 16 . . . . . . 7  |-  ( dom 
Q  =  ( N 
\  { K }
)  ->  dom  ( Q 
\  _I  )  C_  dom  Q )
13 difssd 3591 . . . . . . . 8  |-  ( dom 
Q  =  ( N 
\  { K }
)  ->  ( N  \  { K } ) 
C_  N )
14 sseq1 3484 . . . . . . . 8  |-  ( dom 
Q  =  ( N 
\  { K }
)  ->  ( dom  Q 
C_  N  <->  ( N  \  { K } ) 
C_  N ) )
1513, 14mpbird 232 . . . . . . 7  |-  ( dom 
Q  =  ( N 
\  { K }
)  ->  dom  Q  C_  N )
1612, 15sstrd 3473 . . . . . 6  |-  ( dom 
Q  =  ( N 
\  { K }
)  ->  dom  ( Q 
\  _I  )  C_  N )
178, 9, 163syl 20 . . . . 5  |-  ( Q : ( N  \  { K } ) -1-1-onto-> ( N 
\  { K }
)  ->  dom  ( Q 
\  _I  )  C_  N )
18 id 22 . . . . 5  |-  ( dom  ( Q  \  _I  )  ~~  2o  ->  dom  ( Q  \  _I  )  ~~  2o )
197, 17, 183anim123i 1173 . . . 4  |-  ( ( ( N  \  { K } )  e.  _V  /\  Q : ( N 
\  { K }
)
-1-1-onto-> ( N  \  { K } )  /\  dom  ( Q  \  _I  )  ~~  2o )  ->  ( N  e.  _V  /\  dom  ( Q  \  _I  )  C_  N  /\  dom  ( Q  \  _I  )  ~~  2o ) )
206, 19sylbi 195 . . 3  |-  ( Q  e.  T  ->  ( N  e.  _V  /\  dom  ( Q  \  _I  )  C_  N  /\  dom  ( Q  \  _I  )  ~~  2o ) )
21 eqid 2454 . . . 4  |-  (pmTrsp `  N )  =  (pmTrsp `  N )
2221pmtrmvd 16080 . . 3  |-  ( ( N  e.  _V  /\  dom  ( Q  \  _I  )  C_  N  /\  dom  ( Q  \  _I  )  ~~  2o )  ->  dom  ( ( (pmTrsp `  N ) `  dom  ( Q  \  _I  )
)  \  _I  )  =  dom  ( Q  \  _I  ) )
2320, 22syl 16 . 2  |-  ( Q  e.  T  ->  dom  ( ( (pmTrsp `  N ) `  dom  ( Q  \  _I  )
)  \  _I  )  =  dom  ( Q  \  _I  ) )
243, 23syl5eq 2507 1  |-  ( Q  e.  T  ->  dom  ( S  \  _I  )  =  dom  ( Q  \  _I  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3076    \ cdif 3432    C_ wss 3435   {csn 3984   class class class wbr 4399    _I cid 4738   dom cdm 4947   ran crn 4948   -->wf 5521   -1-1-onto->wf1o 5524   ` cfv 5525   2oc2o 7023    ~~ cen 7416  pmTrspcpmtr 16065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-om 6586  df-1o 7029  df-2o 7030  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-pmtr 16066
This theorem is referenced by:  pmtrdifellem3  16102  pmtrdifellem4  16103
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