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

Theorem tposssxp 6484
Description: The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
Assertion
Ref Expression
tposssxp  |- tpos  F  C_  ( ( `' dom  F  u.  { (/) } )  X.  ran  F )

Proof of Theorem tposssxp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-tpos 6480 . . 3  |- tpos  F  =  ( F  o.  (
x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } ) )
2 cossxp 5393 . . 3  |-  ( F  o.  ( x  e.  ( `' dom  F  u.  { (/) } )  |->  U. `' { x } ) )  C_  ( dom  ( x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } )  X.  ran  F )
31, 2eqsstri 3379 . 2  |- tpos  F  C_  ( dom  ( x  e.  ( `' dom  F  u.  { (/) } )  |->  U. `' { x } )  X.  ran  F )
4 eqid 2437 . . . 4  |-  ( x  e.  ( `' dom  F  u.  { (/) } ) 
|->  U. `' { x } )  =  ( x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } )
54dmmptss 5367 . . 3  |-  dom  (
x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } ) 
C_  ( `' dom  F  u.  { (/) } )
6 xpss1 4985 . . 3  |-  ( dom  ( x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } ) 
C_  ( `' dom  F  u.  { (/) } )  ->  ( dom  (
x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } )  X.  ran  F ) 
C_  ( ( `' dom  F  u.  { (/)
} )  X.  ran  F ) )
75, 6ax-mp 8 . 2  |-  ( dom  ( x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } )  X.  ran  F ) 
C_  ( ( `' dom  F  u.  { (/)
} )  X.  ran  F )
83, 7sstri 3358 1  |- tpos  F  C_  ( ( `' dom  F  u.  { (/) } )  X.  ran  F )
Colors of variables: wff set class
Syntax hints:    u. cun 3319    C_ wss 3321   (/)c0 3629   {csn 3815   U.cuni 4016    e. cmpt 4267    X. cxp 4877   `'ccnv 4878   dom cdm 4879   ran crn 4880    o. ccom 4883  tpos ctpos 6479
This theorem is referenced by:  reltpos  6485  tposexg  6494  wuntpos  8610  catcoppccl  14264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-mpt 4269  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-tpos 6480
  Copyright terms: Public domain W3C validator