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Theorem tposfn 6444
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 6443 . 2  |-  ( F : ( A  X.  B ) --> _V  -> tpos  F : ( B  X.  A ) --> _V )
2 dffn2 5532 . 2  |-  ( F  Fn  ( A  X.  B )  <->  F :
( A  X.  B
) --> _V )
3 dffn2 5532 . 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 2899    X. cxp 4816    Fn wfn 5389   -->wf 5390  tpos ctpos 6414
This theorem is referenced by:  tpossym  6447  funcoppc  13999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-tpos 6415
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