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Theorem tposfo2 6980
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 6979 . . . 4  |-  ( Rel 
A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
21adantrd 466 . . 3  |-  ( Rel 
A  ->  ( ( F  Fn  A  /\  ran  F  =  B )  -> tpos  F  Fn  `' A ) )
3 fndm 5660 . . . . . . . . 9  |-  ( F  Fn  A  ->  dom  F  =  A )
43releqd 4907 . . . . . . . 8  |-  ( F  Fn  A  ->  ( Rel  dom  F  <->  Rel  A ) )
54biimparc 485 . . . . . . 7  |-  ( ( Rel  A  /\  F  Fn  A )  ->  Rel  dom 
F )
6 rntpos 6970 . . . . . . 7  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
75, 6syl 17 . . . . . 6  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ran tpos  F  =  ran  F )
87eqeq1d 2404 . . . . 5  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ( ran tpos  F  =  B  <->  ran  F  =  B ) )
98biimprd 223 . . . 4  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ( ran  F  =  B  ->  ran tpos  F  =  B ) )
109expimpd 601 . . 3  |-  ( Rel 
A  ->  ( ( F  Fn  A  /\  ran  F  =  B )  ->  ran tpos  F  =  B ) )
112, 10jcad 531 . 2  |-  ( Rel 
A  ->  ( ( F  Fn  A  /\  ran  F  =  B )  ->  (tpos  F  Fn  `' A  /\  ran tpos  F  =  B ) ) )
12 df-fo 5574 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
13 df-fo 5574 . 2  |-  (tpos  F : `' A -onto-> B  <->  (tpos  F  Fn  `' A  /\  ran tpos  F  =  B ) )
1411, 12, 133imtr4g 270 1  |-  ( Rel 
A  ->  ( F : A -onto-> B  -> tpos  F : `' A -onto-> B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405   `'ccnv 4821   dom cdm 4822   ran crn 4823   Rel wrel 4827    Fn wfn 5563   -onto->wfo 5566  tpos ctpos 6956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-fo 5574  df-fv 5576  df-tpos 6957
This theorem is referenced by:  tposf2  6981  tposf1o2  6983  tposfo  6984  oppglsm  16984
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