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Theorem tposf2 6274
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 5405 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
2 dffn4 5473 . . . . . . 7  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
31, 2sylib 188 . . . . . 6  |-  ( F : A --> B  ->  F : A -onto-> ran  F
)
4 tposfo2 6273 . . . . . 6  |-  ( Rel 
A  ->  ( F : A -onto-> ran  F  -> tpos  F : `' A -onto-> ran  F ) )
53, 4syl5 28 . . . . 5  |-  ( Rel 
A  ->  ( F : A --> B  -> tpos  F : `' A -onto-> ran  F ) )
65imp 418 . . . 4  |-  ( ( Rel  A  /\  F : A --> B )  -> tpos  F : `' A -onto-> ran  F )
7 fof 5467 . . . 4  |-  (tpos  F : `' A -onto-> ran  F  -> tpos  F : `' A --> ran  F )
86, 7syl 15 . . 3  |-  ( ( Rel  A  /\  F : A --> B )  -> tpos  F : `' A --> ran  F
)
9 frn 5411 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
109adantl 452 . . 3  |-  ( ( Rel  A  /\  F : A --> B )  ->  ran  F  C_  B )
11 fss 5413 . . 3  |-  ( (tpos 
F : `' A --> ran  F  /\  ran  F  C_  B )  -> tpos  F : `' A --> B )
128, 10, 11syl2anc 642 . 2  |-  ( ( Rel  A  /\  F : A --> B )  -> tpos  F : `' A --> B )
1312ex 423 1  |-  ( Rel 
A  ->  ( F : A --> B  -> tpos  F : `' A --> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    C_ wss 3165   `'ccnv 4704   ran crn 4706   Rel wrel 4710    Fn wfn 5266   -->wf 5267   -onto->wfo 5269  tpos ctpos 6249
This theorem is referenced by:  tposf  6278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-tpos 6250
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