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Theorem tposfn2 6977
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 6971 . . . 4  |-  ( Fun 
F  ->  Fun tpos  F )
21a1i 11 . . 3  |-  ( Rel 
A  ->  ( Fun  F  ->  Fun tpos  F )
)
3 dmtpos 6967 . . . . . 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 5085 . . . . 5  |-  ( dom 
F  =  A  -> 
( Rel  dom  F  <->  Rel  A ) )
6 cnveq 5176 . . . . . 6  |-  ( dom 
F  =  A  ->  `' dom  F  =  `' A )
76eqeq2d 2481 . . . . 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 5591 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
12 df-fn 5591 . 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 1379   `'ccnv 4998   dom cdm 4999   Rel wrel 5004   Fun wfun 5582    Fn wfn 5583  tpos ctpos 6954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-tpos 6955
This theorem is referenced by:  tposfo2  6978  tpos0  6985
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