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Theorem pmtrdifellem2 16701
Description: Lemma 2 for pmtrdifel 16704. (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 3604 . . 3  |-  ( S 
\  _I  )  =  ( ( (pmTrsp `  N ) `  dom  ( Q  \  _I  )
)  \  _I  )
32dmeqi 5193 . 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 16689 . . . 4  |-  ( Q  e.  T  <->  ( ( N  \  { K }
)  e.  _V  /\  Q : ( N  \  { K } ) -1-1-onto-> ( N 
\  { K }
)  /\  dom  ( Q 
\  _I  )  ~~  2o ) )
7 difsnexi 6581 . . . . 5  |-  ( ( N  \  { K } )  e.  _V  ->  N  e.  _V )
8 f1of 5798 . . . . . 6  |-  ( Q : ( N  \  { K } ) -1-1-onto-> ( N 
\  { K }
)  ->  Q :
( N  \  { K } ) --> ( N 
\  { K }
) )
9 fdm 5717 . . . . . 6  |-  ( Q : ( N  \  { K } ) --> ( N  \  { K } )  ->  dom  Q  =  ( N  \  { K } ) )
10 difssd 3618 . . . . . . . 8  |-  ( dom 
Q  =  ( N 
\  { K }
)  ->  ( Q  \  _I  )  C_  Q
)
11 dmss 5191 . . . . . . . 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 3618 . . . . . . . 8  |-  ( dom 
Q  =  ( N 
\  { K }
)  ->  ( N  \  { K } ) 
C_  N )
14 sseq1 3510 . . . . . . . 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 3499 . . . . . 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 1179 . . . 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 16680 . . 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 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    \ cdif 3458    C_ wss 3461   {csn 4016   class class class wbr 4439    _I cid 4779   dom cdm 4988   ran crn 4989   -->wf 5566   -1-1-onto->wf1o 5569   ` cfv 5570   2oc2o 7116    ~~ cen 7506  pmTrspcpmtr 16665
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-3or 972  df-3an 973  df-tru 1401  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-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  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-om 6674  df-1o 7122  df-2o 7123  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pmtr 16666
This theorem is referenced by:  pmtrdifellem3  16702  pmtrdifellem4  16703
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