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Theorem tposfn2 6264
Description: The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposfn2  |-  ( Rel 
A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )

Proof of Theorem tposfn2
StepHypRef Expression
1 tposfun 6258 . . . 4  |-  ( Fun 
F  ->  Fun tpos  F )
21a1i 9 . . 3  |-  ( Rel 
A  ->  ( Fun  F  ->  Fun tpos  F )
)
3 dmtpos 6254 . . . . . 6  |-  ( Rel 
dom  F  ->  dom tpos  F  =  `' dom  F )
43a1i 9 . . . . 5  |-  ( dom 
F  =  A  -> 
( Rel  dom  F  ->  dom tpos  F  =  `' dom  F ) )
5 releq 4707 . . . . 5  |-  ( dom 
F  =  A  -> 
( Rel  dom  F  <->  Rel  A ) )
6 cnveq 4800 . . . . . 6  |-  ( dom 
F  =  A  ->  `' dom  F  =  `' A )
76eqeq2d 2189 . . . . 5  |-  ( dom 
F  =  A  -> 
( dom tpos  F  =  `' dom  F  <->  dom tpos  F  =  `' A ) )
84, 5, 73imtr3d 202 . . . 4  |-  ( dom 
F  =  A  -> 
( Rel  A  ->  dom tpos  F  =  `' A
) )
98com12 30 . . 3  |-  ( Rel 
A  ->  ( dom  F  =  A  ->  dom tpos  F  =  `' A ) )
102, 9anim12d 335 . 2  |-  ( Rel 
A  ->  ( ( Fun  F  /\  dom  F  =  A )  ->  ( Fun tpos  F  /\  dom tpos  F  =  `' A ) ) )
11 df-fn 5218 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
12 df-fn 5218 . 2  |-  (tpos  F  Fn  `' A  <->  ( Fun tpos  F  /\  dom tpos  F  =  `' A
) )
1310, 11, 123imtr4g 205 1  |-  ( Rel 
A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353   `'ccnv 4624   dom cdm 4625   Rel wrel 4630   Fun wfun 5209    Fn wfn 5210  tpos ctpos 6242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-tpos 6243
This theorem is referenced by:  tposfo2  6265  tpos0  6272
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