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Theorem tposfo2 6273
Description: Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposfo2  |-  ( Rel 
A  ->  ( F : A -onto-> B  -> tpos  F : `' A -onto-> B ) )

Proof of Theorem tposfo2
StepHypRef Expression
1 tposfn2 6272 . . . 4  |-  ( Rel 
A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
21adantrd 454 . . 3  |-  ( Rel 
A  ->  ( ( F  Fn  A  /\  ran  F  =  B )  -> tpos  F  Fn  `' A ) )
3 fndm 5359 . . . . . . . . 9  |-  ( F  Fn  A  ->  dom  F  =  A )
43releqd 4789 . . . . . . . 8  |-  ( F  Fn  A  ->  ( Rel  dom  F  <->  Rel  A ) )
54biimparc 473 . . . . . . 7  |-  ( ( Rel  A  /\  F  Fn  A )  ->  Rel  dom 
F )
6 rntpos 6263 . . . . . . 7  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
75, 6syl 15 . . . . . 6  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ran tpos  F  =  ran  F )
87eqeq1d 2304 . . . . 5  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ( ran tpos  F  =  B  <->  ran  F  =  B ) )
98biimprd 214 . . . 4  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ( ran  F  =  B  ->  ran tpos  F  =  B ) )
109expimpd 586 . . 3  |-  ( Rel 
A  ->  ( ( F  Fn  A  /\  ran  F  =  B )  ->  ran tpos  F  =  B ) )
112, 10jcad 519 . 2  |-  ( Rel 
A  ->  ( ( F  Fn  A  /\  ran  F  =  B )  ->  (tpos  F  Fn  `' A  /\  ran tpos  F  =  B ) ) )
12 df-fo 5277 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
13 df-fo 5277 . 2  |-  (tpos  F : `' A -onto-> B  <->  (tpos  F  Fn  `' A  /\  ran tpos  F  =  B ) )
1411, 12, 133imtr4g 261 1  |-  ( Rel 
A  ->  ( F : A -onto-> B  -> tpos  F : `' A -onto-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632   `'ccnv 4704   dom cdm 4705   ran crn 4706   Rel wrel 4710    Fn wfn 5266   -onto->wfo 5269  tpos ctpos 6249
This theorem is referenced by:  tposf2  6274  tposf1o2  6276  tposfo  6277  oppglsm  14969
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-fo 5277  df-fv 5279  df-tpos 6250
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