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Theorem tposf2 7005
Description: The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposf2  |-  ( Rel 
A  ->  ( F : A --> B  -> tpos  F : `' A --> B ) )

Proof of Theorem tposf2
StepHypRef Expression
1 ffn 5746 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
2 dffn4 5816 . . . . . . 7  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
31, 2sylib 199 . . . . . 6  |-  ( F : A --> B  ->  F : A -onto-> ran  F
)
4 tposfo2 7004 . . . . . 6  |-  ( Rel 
A  ->  ( F : A -onto-> ran  F  -> tpos  F : `' A -onto-> ran  F ) )
53, 4syl5 33 . . . . 5  |-  ( Rel 
A  ->  ( F : A --> B  -> tpos  F : `' A -onto-> ran  F ) )
65imp 430 . . . 4  |-  ( ( Rel  A  /\  F : A --> B )  -> tpos  F : `' A -onto-> ran  F )
7 fof 5810 . . . 4  |-  (tpos  F : `' A -onto-> ran  F  -> tpos  F : `' A --> ran  F )
86, 7syl 17 . . 3  |-  ( ( Rel  A  /\  F : A --> B )  -> tpos  F : `' A --> ran  F
)
9 frn 5752 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
109adantl 467 . . 3  |-  ( ( Rel  A  /\  F : A --> B )  ->  ran  F  C_  B )
118, 10fssd 5755 . 2  |-  ( ( Rel  A  /\  F : A --> B )  -> tpos  F : `' A --> B )
1211ex 435 1  |-  ( Rel 
A  ->  ( F : A --> B  -> tpos  F : `' A --> B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    C_ wss 3442   `'ccnv 4853   ran crn 4855   Rel wrel 4859    Fn wfn 5596   -->wf 5597   -onto->wfo 5599  tpos ctpos 6980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-tpos 6981
This theorem is referenced by:  tposf  7009
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