MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpos0 Structured version   Unicode version

Theorem tpos0 7011
Description: Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tpos0  |- tpos  (/)  =  (/)

Proof of Theorem tpos0
StepHypRef Expression
1 rel0 4978 . . . 4  |-  Rel  (/)
2 eqid 2429 . . . . 5  |-  (/)  =  (/)
3 fn0 5713 . . . . 5  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
42, 3mpbir 212 . . . 4  |-  (/)  Fn  (/)
5 tposfn2 7003 . . . 4  |-  ( Rel  (/)  ->  ( (/)  Fn  (/)  -> tpos  (/)  Fn  `' (/) ) )
61, 4, 5mp2 9 . . 3  |- tpos  (/)  Fn  `' (/)
7 cnv0 5259 . . . 4  |-  `' (/)  =  (/)
87fneq2i 5689 . . 3  |-  (tpos  (/)  Fn  `' (/)  <-> tpos  (/)  Fn  (/) )
96, 8mpbi 211 . 2  |- tpos  (/)  Fn  (/)
10 fn0 5713 . 2  |-  (tpos  (/)  Fn  (/)  <-> tpos  (/)  =  (/) )
119, 10mpbi 211 1  |- tpos  (/)  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   (/)c0 3767   `'ccnv 4853   Rel wrel 4859    Fn wfn 5596  tpos ctpos 6980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-tpos 6981
This theorem is referenced by:  oppchomfval  15570  oppgplusfval  16950  opprmulfval  17788
  Copyright terms: Public domain W3C validator