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Theorem tposssxp 6951
Description: The transposition is a subset of a Cartesian 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 6947 . . 3  |- tpos  F  =  ( F  o.  (
x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } ) )
2 cossxp 5513 . . 3  |-  ( F  o.  ( x  e.  ( `' dom  F  u.  { (/) } )  |->  U. `' { x } ) )  C_  ( dom  ( x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } )  X.  ran  F )
31, 2eqsstri 3519 . 2  |- tpos  F  C_  ( dom  ( x  e.  ( `' dom  F  u.  { (/) } )  |->  U. `' { x } )  X.  ran  F )
4 eqid 2454 . . . 4  |-  ( x  e.  ( `' dom  F  u.  { (/) } ) 
|->  U. `' { x } )  =  ( x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } )
54dmmptss 5486 . . 3  |-  dom  (
x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } ) 
C_  ( `' dom  F  u.  { (/) } )
6 xpss1 5099 . . 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 5 . 2  |-  ( dom  ( x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } )  X.  ran  F ) 
C_  ( ( `' dom  F  u.  { (/)
} )  X.  ran  F )
83, 7sstri 3498 1  |- tpos  F  C_  ( ( `' dom  F  u.  { (/) } )  X.  ran  F )
Colors of variables: wff setvar class
Syntax hints:    u. cun 3459    C_ wss 3461   (/)c0 3783   {csn 4016   U.cuni 4235    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   dom cdm 4988   ran crn 4989    o. ccom 4992  tpos ctpos 6946
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-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-mpt 4499  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-tpos 6947
This theorem is referenced by:  reltpos  6952  tposexg  6961  wuntpos  9101  catcoppccl  15589
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