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Theorem tposf2 5938
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 5098 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
2 dffn4 5164 . . . . . . 7  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
31, 2sylib 120 . . . . . 6  |-  ( F : A --> B  ->  F : A -onto-> ran  F
)
4 tposfo2 5937 . . . . . 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 122 . . . 4  |-  ( ( Rel  A  /\  F : A --> B )  -> tpos  F : `' A -onto-> ran  F )
7 fof 5158 . . . 4  |-  (tpos  F : `' A -onto-> ran  F  -> tpos  F : `' A --> ran  F )
86, 7syl 14 . . 3  |-  ( ( Rel  A  /\  F : A --> B )  -> tpos  F : `' A --> ran  F
)
9 frn 5104 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
109adantl 271 . . 3  |-  ( ( Rel  A  /\  F : A --> B )  ->  ran  F  C_  B )
11 fss 5106 . . 3  |-  ( (tpos 
F : `' A --> ran  F  /\  ran  F  C_  B )  -> tpos  F : `' A --> B )
128, 10, 11syl2anc 403 . 2  |-  ( ( Rel  A  /\  F : A --> B )  -> tpos  F : `' A --> B )
1312ex 113 1  |-  ( Rel 
A  ->  ( F : A --> B  -> tpos  F : `' A --> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    C_ wss 2983   `'ccnv 4391   ran crn 4393   Rel wrel 4397    Fn wfn 4948   -->wf 4949   -onto->wfo 4951  tpos ctpos 5914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-fo 4959  df-fv 4961  df-tpos 5915
This theorem is referenced by:  tposf  5942
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