MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pmtrdifellem2 Structured version   Unicode version

Theorem pmtrdifellem2 15974
Description: Lemma 2 for pmtrdifel 15977. (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 3465 . . 3  |-  ( S 
\  _I  )  =  ( ( (pmTrsp `  N ) `  dom  ( Q  \  _I  )
)  \  _I  )
32dmeqi 5036 . 2  |-  dom  ( S  \  _I  )  =  dom  ( ( (pmTrsp `  N ) `  dom  ( Q  \  _I  )
)  \  _I  )
4 eqid 2438 . . . . 5  |-  (pmTrsp `  ( N  \  { K } ) )  =  (pmTrsp `  ( N  \  { K } ) )
5 pmtrdifel.t . . . . 5  |-  T  =  ran  (pmTrsp `  ( N  \  { K }
) )
64, 5pmtrfb 15962 . . . 4  |-  ( Q  e.  T  <->  ( ( N  \  { K }
)  e.  _V  /\  Q : ( N  \  { K } ) -1-1-onto-> ( N 
\  { K }
)  /\  dom  ( Q 
\  _I  )  ~~  2o ) )
7 difsnexi 6379 . . . . 5  |-  ( ( N  \  { K } )  e.  _V  ->  N  e.  _V )
8 f1of 5636 . . . . . 6  |-  ( Q : ( N  \  { K } ) -1-1-onto-> ( N 
\  { K }
)  ->  Q :
( N  \  { K } ) --> ( N 
\  { K }
) )
9 fdm 5558 . . . . . 6  |-  ( Q : ( N  \  { K } ) --> ( N  \  { K } )  ->  dom  Q  =  ( N  \  { K } ) )
10 difssd 3479 . . . . . . . 8  |-  ( dom 
Q  =  ( N 
\  { K }
)  ->  ( Q  \  _I  )  C_  Q
)
11 dmss 5034 . . . . . . . 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 3479 . . . . . . . 8  |-  ( dom 
Q  =  ( N 
\  { K }
)  ->  ( N  \  { K } ) 
C_  N )
14 sseq1 3372 . . . . . . . 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 3361 . . . . . 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 2438 . . . 4  |-  (pmTrsp `  N )  =  (pmTrsp `  N )
2221pmtrmvd 15953 . . 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 2482 1  |-  ( Q  e.  T  ->  dom  ( S  \  _I  )  =  dom  ( Q  \  _I  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2967    \ cdif 3320    C_ wss 3323   {csn 3872   class class class wbr 4287    _I cid 4626   dom cdm 4835   ran crn 4836   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413   2oc2o 6906    ~~ cen 7299  pmTrspcpmtr 15938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-1o 6912  df-2o 6913  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pmtr 15939
This theorem is referenced by:  pmtrdifellem3  15975  pmtrdifellem4  15976
  Copyright terms: Public domain W3C validator