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Theorem tposf2 6258
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 5389 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
2 dffn4 5457 . . . . . . 7  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
31, 2sylib 188 . . . . . 6  |-  ( F : A --> B  ->  F : A -onto-> ran  F
)
4 tposfo2 6257 . . . . . 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 5451 . . . 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 5395 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
109adantl 452 . . 3  |-  ( ( Rel  A  /\  F : A --> B )  ->  ran  F  C_  B )
11 fss 5397 . . 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 3152   `'ccnv 4688   ran crn 4690   Rel wrel 4694    Fn wfn 5250   -->wf 5251   -onto->wfo 5253  tpos ctpos 6233
This theorem is referenced by:  tposf  6262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-tpos 6234
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