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Theorem tposfn2 6765
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 6759 . . . 4  |-  ( Fun 
F  ->  Fun tpos  F )
21a1i 11 . . 3  |-  ( Rel 
A  ->  ( Fun  F  ->  Fun tpos  F )
)
3 dmtpos 6755 . . . . . 6  |-  ( Rel 
dom  F  ->  dom tpos  F  =  `' dom  F )
43a1i 11 . . . . 5  |-  ( dom 
F  =  A  -> 
( Rel  dom  F  ->  dom tpos  F  =  `' dom  F ) )
5 releq 4920 . . . . 5  |-  ( dom 
F  =  A  -> 
( Rel  dom  F  <->  Rel  A ) )
6 cnveq 5011 . . . . . 6  |-  ( dom 
F  =  A  ->  `' dom  F  =  `' A )
76eqeq2d 2452 . . . . 5  |-  ( dom 
F  =  A  -> 
( dom tpos  F  =  `' dom  F  <->  dom tpos  F  =  `' A ) )
84, 5, 73imtr3d 267 . . . 4  |-  ( dom 
F  =  A  -> 
( Rel  A  ->  dom tpos  F  =  `' A
) )
98com12 31 . . 3  |-  ( Rel 
A  ->  ( dom  F  =  A  ->  dom tpos  F  =  `' A ) )
102, 9anim12d 563 . 2  |-  ( Rel 
A  ->  ( ( Fun  F  /\  dom  F  =  A )  ->  ( Fun tpos  F  /\  dom tpos  F  =  `' A ) ) )
11 df-fn 5419 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
12 df-fn 5419 . 2  |-  (tpos  F  Fn  `' A  <->  ( Fun tpos  F  /\  dom tpos  F  =  `' A
) )
1310, 11, 123imtr4g 270 1  |-  ( Rel 
A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   `'ccnv 4837   dom cdm 4838   Rel wrel 4843   Fun wfun 5410    Fn wfn 5411  tpos ctpos 6742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-fv 5424  df-tpos 6743
This theorem is referenced by:  tposfo2  6766  tpos0  6773
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