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Theorem tpos0 6982
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 5125 . . . 4  |-  Rel  (/)
2 eqid 2467 . . . . 5  |-  (/)  =  (/)
3 fn0 5698 . . . . 5  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
42, 3mpbir 209 . . . 4  |-  (/)  Fn  (/)
5 tposfn2 6974 . . . 4  |-  ( Rel  (/)  ->  ( (/)  Fn  (/)  -> tpos  (/)  Fn  `' (/) ) )
61, 4, 5mp2 9 . . 3  |- tpos  (/)  Fn  `' (/)
7 cnv0 5407 . . . 4  |-  `' (/)  =  (/)
87fneq2i 5674 . . 3  |-  (tpos  (/)  Fn  `' (/)  <-> tpos  (/)  Fn  (/) )
96, 8mpbi 208 . 2  |- tpos  (/)  Fn  (/)
10 fn0 5698 . 2  |-  (tpos  (/)  Fn  (/)  <-> tpos  (/)  =  (/) )
119, 10mpbi 208 1  |- tpos  (/)  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   (/)c0 3785   `'ccnv 4998   Rel wrel 5004    Fn wfn 5581  tpos ctpos 6951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-tpos 6952
This theorem is referenced by:  oppchomfval  14963  oppgplusfval  16175  opprmulfval  17055
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