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

Theorem tposfn 6500
Description: Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
tposfn  |-  ( F  Fn  ( A  X.  B )  -> tpos  F  Fn  ( B  X.  A
) )

Proof of Theorem tposfn
StepHypRef Expression
1 tposf 6499 . 2  |-  ( F : ( A  X.  B ) --> _V  -> tpos  F : ( B  X.  A ) --> _V )
2 dffn2 5584 . 2  |-  ( F  Fn  ( A  X.  B )  <->  F :
( A  X.  B
) --> _V )
3 dffn2 5584 . 2  |-  (tpos  F  Fn  ( B  X.  A
)  <-> tpos  F : ( B  X.  A ) --> _V )
41, 2, 33imtr4i 258 1  |-  ( F  Fn  ( A  X.  B )  -> tpos  F  Fn  ( B  X.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2948    X. cxp 4868    Fn wfn 5441   -->wf 5442  tpos ctpos 6470
This theorem is referenced by:  tpossym  6503  funcoppc  14064
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-13 1727  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-pow 4369  ax-pr 4395  ax-un 4693
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-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  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-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-tpos 6471
  Copyright terms: Public domain W3C validator