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Theorem tpos0 6512
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 5002 . . . 4  |-  Rel  (/)
2 eqid 2438 . . . . 5  |-  (/)  =  (/)
3 fn0 5567 . . . . 5  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
42, 3mpbir 202 . . . 4  |-  (/)  Fn  (/)
5 tposfn2 6504 . . . 4  |-  ( Rel  (/)  ->  ( (/)  Fn  (/)  -> tpos  (/)  Fn  `' (/) ) )
61, 4, 5mp2 9 . . 3  |- tpos  (/)  Fn  `' (/)
7 cnv0 5278 . . . 4  |-  `' (/)  =  (/)
87fneq2i 5543 . . 3  |-  (tpos  (/)  Fn  `' (/)  <-> tpos  (/)  Fn  (/) )
96, 8mpbi 201 . 2  |- tpos  (/)  Fn  (/)
10 fn0 5567 . 2  |-  (tpos  (/)  Fn  (/)  <-> tpos  (/)  =  (/) )
119, 10mpbi 201 1  |- tpos  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1653   (/)c0 3630   `'ccnv 4880   Rel wrel 4886    Fn wfn 5452  tpos ctpos 6481
This theorem is referenced by:  oppchomfval  13945  oppgplusfval  15149  opprmulfval  15735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465  df-tpos 6482
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