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Theorem tposfn2 6493
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 6487 . . . 4  |-  ( Fun 
F  ->  Fun tpos  F )
21a1i 11 . . 3  |-  ( Rel 
A  ->  ( Fun  F  ->  Fun tpos  F )
)
3 dmtpos 6483 . . . . . 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 4951 . . . . 5  |-  ( dom 
F  =  A  -> 
( Rel  dom  F  <->  Rel  A ) )
6 cnveq 5038 . . . . . 6  |-  ( dom 
F  =  A  ->  `' dom  F  =  `' A )
76eqeq2d 2446 . . . . 5  |-  ( dom 
F  =  A  -> 
( dom tpos  F  =  `' dom  F  <->  dom tpos  F  =  `' A ) )
84, 5, 73imtr3d 259 . . . 4  |-  ( dom 
F  =  A  -> 
( Rel  A  ->  dom tpos  F  =  `' A
) )
98com12 29 . . 3  |-  ( Rel 
A  ->  ( dom  F  =  A  ->  dom tpos  F  =  `' A ) )
102, 9anim12d 547 . 2  |-  ( Rel 
A  ->  ( ( Fun  F  /\  dom  F  =  A )  ->  ( Fun tpos  F  /\  dom tpos  F  =  `' A ) ) )
11 df-fn 5449 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
12 df-fn 5449 . 2  |-  (tpos  F  Fn  `' A  <->  ( Fun tpos  F  /\  dom tpos  F  =  `' A
) )
1310, 11, 123imtr4g 262 1  |-  ( Rel 
A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652   `'ccnv 4869   dom cdm 4870   Rel wrel 4875   Fun wfun 5440    Fn wfn 5441  tpos ctpos 6470
This theorem is referenced by:  tposfo2  6494  tpos0  6501
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-fv 5454  df-tpos 6471
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