MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tposfo2 Unicode version

Theorem tposfo2 6257
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 6256 . . . 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 5343 . . . . . . . . 9  |-  ( F  Fn  A  ->  dom  F  =  A )
43releqd 4773 . . . . . . . 8  |-  ( F  Fn  A  ->  ( Rel  dom  F  <->  Rel  A ) )
54biimparc 473 . . . . . . 7  |-  ( ( Rel  A  /\  F  Fn  A )  ->  Rel  dom 
F )
6 rntpos 6247 . . . . . . 7  |-  ( Rel 
dom  F  ->  ran tpos  F  =  ran  F )
75, 6syl 15 . . . . . 6  |-  ( ( Rel  A  /\  F  Fn  A )  ->  ran tpos  F  =  ran  F )
87eqeq1d 2291 . . . . 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 5261 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
13 df-fo 5261 . 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 1623   `'ccnv 4688   dom cdm 4689   ran crn 4690   Rel wrel 4694    Fn wfn 5250   -onto->wfo 5253  tpos ctpos 6233
This theorem is referenced by:  tposf2  6258  tposf1o2  6260  tposfo  6261  oppglsm  14953
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-fo 5261  df-fv 5263  df-tpos 6234
  Copyright terms: Public domain W3C validator