ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tposfo2 Structured version   Unicode version

Theorem tposfo2 5823
Description: Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposfo2  Rel  F : -onto-> tpos  F : `' -onto->

Proof of Theorem tposfo2
StepHypRef Expression
1 tposfn2 5822 . . . 4  Rel  F  Fn tpos  F  Fn  `'
21adantrd 264 . . 3  Rel  F  Fn  ran  F tpos  F  Fn  `'
3 fndm 4941 . . . . . . . . 9  F  Fn  dom  F
43releqd 4367 . . . . . . . 8  F  Fn  Rel  dom  F  Rel
54biimparc 283 . . . . . . 7  Rel  F  Fn  Rel  dom 
F
6 rntpos 5813 . . . . . . 7  Rel 
dom  F  ran tpos  F 
ran  F
75, 6syl 14 . . . . . 6  Rel  F  Fn  ran tpos  F  ran  F
87eqeq1d 2045 . . . . 5  Rel  F  Fn  ran tpos  F  ran  F
98biimprd 147 . . . 4  Rel  F  Fn  ran  F  ran tpos  F
109expimpd 345 . . 3  Rel  F  Fn  ran  F 
ran tpos  F
112, 10jcad 291 . 2  Rel  F  Fn  ran  F tpos  F  Fn  `'  ran tpos  F
12 df-fo 4851 . 2  F : -onto->  F  Fn  ran  F
13 df-fo 4851 . 2 tpos  F : `' -onto-> tpos  F  Fn  `'  ran tpos  F
1411, 12, 133imtr4g 194 1  Rel  F : -onto-> tpos  F : `' -onto->
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   `'ccnv 4287   dom cdm 4288   ran crn 4289   Rel wrel 4293    Fn wfn 4840   -onto->wfo 4843  tpos ctpos 5800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fo 4851  df-fv 4853  df-tpos 5801
This theorem is referenced by:  tposf2  5824  tposf1o2  5826  tposfo  5827
  Copyright terms: Public domain W3C validator