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Theorem tposfn2 6439
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 6433 . . . 4  |-  ( Fun 
F  ->  Fun tpos  F )
21a1i 11 . . 3  |-  ( Rel 
A  ->  ( Fun  F  ->  Fun tpos  F )
)
3 dmtpos 6429 . . . . . 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 4901 . . . . 5  |-  ( dom 
F  =  A  -> 
( Rel  dom  F  <->  Rel  A ) )
6 cnveq 4988 . . . . . 6  |-  ( dom 
F  =  A  ->  `' dom  F  =  `' A )
76eqeq2d 2400 . . . . 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 5399 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
12 df-fn 5399 . 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 1649   `'ccnv 4819   dom cdm 4820   Rel wrel 4825   Fun wfun 5390    Fn wfn 5391  tpos ctpos 6416
This theorem is referenced by:  tposfo2  6440  tpos0  6447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-fv 5404  df-tpos 6417
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