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

Theorem tposfun 6487
Description: The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
tposfun  |-  ( Fun 
F  ->  Fun tpos  F )

Proof of Theorem tposfun
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funmpt 5481 . . 3  |-  Fun  (
x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } )
2 funco 5483 . . 3  |-  ( ( Fun  F  /\  Fun  ( x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } ) )  ->  Fun  ( F  o.  ( x  e.  ( `' dom  F  u.  { (/) } )  |->  U. `' { x } ) ) )
31, 2mpan2 653 . 2  |-  ( Fun 
F  ->  Fun  ( F  o.  ( x  e.  ( `' dom  F  u.  { (/) } )  |->  U. `' { x } ) ) )
4 df-tpos 6471 . . 3  |- tpos  F  =  ( F  o.  (
x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } ) )
54funeqi 5466 . 2  |-  ( Fun tpos  F 
<->  Fun  ( F  o.  ( x  e.  ( `' dom  F  u.  { (/)
} )  |->  U. `' { x } ) ) )
63, 5sylibr 204 1  |-  ( Fun 
F  ->  Fun tpos  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    u. cun 3310   (/)c0 3620   {csn 3806   U.cuni 4007    e. cmpt 4258   `'ccnv 4869   dom cdm 4870    o. ccom 4874   Fun wfun 5440  tpos ctpos 6470
This theorem is referenced by:  tposfn2  6493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-fun 5448  df-tpos 6471
  Copyright terms: Public domain W3C validator