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Theorem tposfo2 6118
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 6117 . . . 4  |-  ( Rel 
A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
21adantrd 275 . . 3  |-  ( Rel 
A  ->  ( ( F  Fn  A  /\  ran  F  =  B )  -> tpos  F  Fn  `' A ) )
3 fndm 5180 . . . . . . . . 9  |-  ( F  Fn  A  ->  dom  F  =  A )
43releqd 4583 . . . . . . . 8  |-  ( F  Fn  A  ->  ( Rel  dom  F  <->  Rel  A ) )
54biimparc 295 . . . . . . 7  |-  ( ( Rel  A  /\  F  Fn  A )  ->  Rel  dom 
F )
6 rntpos 6108 . . . . . . 7  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
75, 6syl 14 . . . . . 6  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ran tpos  F  =  ran  F )
87eqeq1d 2123 . . . . 5  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ( ran tpos  F  =  B  <->  ran  F  =  B ) )
98biimprd 157 . . . 4  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ( ran  F  =  B  ->  ran tpos  F  =  B ) )
109expimpd 358 . . 3  |-  ( Rel 
A  ->  ( ( F  Fn  A  /\  ran  F  =  B )  ->  ran tpos  F  =  B ) )
112, 10jcad 303 . 2  |-  ( Rel 
A  ->  ( ( F  Fn  A  /\  ran  F  =  B )  ->  (tpos  F  Fn  `' A  /\  ran tpos  F  =  B ) ) )
12 df-fo 5087 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
13 df-fo 5087 . 2  |-  (tpos  F : `' A -onto-> B  <->  (tpos  F  Fn  `' A  /\  ran tpos  F  =  B ) )
1411, 12, 133imtr4g 204 1  |-  ( Rel 
A  ->  ( F : A -onto-> B  -> tpos  F : `' A -onto-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314   `'ccnv 4498   dom cdm 4499   ran crn 4500   Rel wrel 4504    Fn wfn 5076   -onto->wfo 5079  tpos ctpos 6095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-fo 5087  df-fv 5089  df-tpos 6096
This theorem is referenced by:  tposf2  6119  tposf1o2  6121  tposfo  6122
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