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Theorem tposf2 6980
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 5731 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
2 dffn4 5801 . . . . . . 7  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
31, 2sylib 196 . . . . . 6  |-  ( F : A --> B  ->  F : A -onto-> ran  F
)
4 tposfo2 6979 . . . . . 6  |-  ( Rel 
A  ->  ( F : A -onto-> ran  F  -> tpos  F : `' A -onto-> ran  F ) )
53, 4syl5 32 . . . . 5  |-  ( Rel 
A  ->  ( F : A --> B  -> tpos  F : `' A -onto-> ran  F ) )
65imp 429 . . . 4  |-  ( ( Rel  A  /\  F : A --> B )  -> tpos  F : `' A -onto-> ran  F )
7 fof 5795 . . . 4  |-  (tpos  F : `' A -onto-> ran  F  -> tpos  F : `' A --> ran  F )
86, 7syl 16 . . 3  |-  ( ( Rel  A  /\  F : A --> B )  -> tpos  F : `' A --> ran  F
)
9 frn 5737 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
109adantl 466 . . 3  |-  ( ( Rel  A  /\  F : A --> B )  ->  ran  F  C_  B )
11 fss 5739 . . 3  |-  ( (tpos 
F : `' A --> ran  F  /\  ran  F  C_  B )  -> tpos  F : `' A --> B )
128, 10, 11syl2anc 661 . 2  |-  ( ( Rel  A  /\  F : A --> B )  -> tpos  F : `' A --> B )
1312ex 434 1  |-  ( Rel 
A  ->  ( F : A --> B  -> tpos  F : `' A --> B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    C_ wss 3476   `'ccnv 4998   ran crn 5000   Rel wrel 5004    Fn wfn 5583   -->wf 5584   -onto->wfo 5586  tpos ctpos 6955
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 6577
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-f 5592  df-fo 5594  df-fv 5596  df-tpos 6956
This theorem is referenced by:  tposf  6984
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