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Theorem tposfn 5888
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 5887 . 2  |-  ( F : ( A  X.  B ) --> _V  -> tpos  F : ( B  X.  A ) --> _V )
2 dffn2 5047 . 2  |-  ( F  Fn  ( A  X.  B )  <->  F :
( A  X.  B
) --> _V )
3 dffn2 5047 . 2  |-  (tpos  F  Fn  ( B  X.  A
)  <-> tpos  F : ( B  X.  A ) --> _V )
41, 2, 33imtr4i 190 1  |-  ( F  Fn  ( A  X.  B )  -> tpos  F  Fn  ( B  X.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2557    X. cxp 4343    Fn wfn 4897   -->wf 4898  tpos ctpos 5859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-tpos 5860
This theorem is referenced by:  tpossym  5891
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