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Theorem tpos0 6502
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 4992 . . . 4  |-  Rel  (/)
2 eqid 2436 . . . . 5  |-  (/)  =  (/)
3 fn0 5557 . . . . 5  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
42, 3mpbir 201 . . . 4  |-  (/)  Fn  (/)
5 tposfn2 6494 . . . 4  |-  ( Rel  (/)  ->  ( (/)  Fn  (/)  -> tpos  (/)  Fn  `' (/) ) )
61, 4, 5mp2 9 . . 3  |- tpos  (/)  Fn  `' (/)
7 cnv0 5268 . . . 4  |-  `' (/)  =  (/)
87fneq2i 5533 . . 3  |-  (tpos  (/)  Fn  `' (/)  <-> tpos  (/)  Fn  (/) )
96, 8mpbi 200 . 2  |- tpos  (/)  Fn  (/)
10 fn0 5557 . 2  |-  (tpos  (/)  Fn  (/)  <-> tpos  (/)  =  (/) )
119, 10mpbi 200 1  |- tpos  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1652   (/)c0 3621   `'ccnv 4870   Rel wrel 4876    Fn wfn 5442  tpos ctpos 6471
This theorem is referenced by:  oppchomfval  13933  oppgplusfval  15137  opprmulfval  15723
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-fv 5455  df-tpos 6472
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