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Theorem tpos0 6888
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 5075 . . . 4  |-  Rel  (/)
2 eqid 2454 . . . . 5  |-  (/)  =  (/)
3 fn0 5641 . . . . 5  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
42, 3mpbir 209 . . . 4  |-  (/)  Fn  (/)
5 tposfn2 6880 . . . 4  |-  ( Rel  (/)  ->  ( (/)  Fn  (/)  -> tpos  (/)  Fn  `' (/) ) )
61, 4, 5mp2 9 . . 3  |- tpos  (/)  Fn  `' (/)
7 cnv0 5351 . . . 4  |-  `' (/)  =  (/)
87fneq2i 5617 . . 3  |-  (tpos  (/)  Fn  `' (/)  <-> tpos  (/)  Fn  (/) )
96, 8mpbi 208 . 2  |- tpos  (/)  Fn  (/)
10 fn0 5641 . 2  |-  (tpos  (/)  Fn  (/)  <-> tpos  (/)  =  (/) )
119, 10mpbi 208 1  |- tpos  (/)  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   (/)c0 3748   `'ccnv 4950   Rel wrel 4956    Fn wfn 5524  tpos ctpos 6857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-fv 5537  df-tpos 6858
This theorem is referenced by:  oppchomfval  14775  oppgplusfval  15985  opprmulfval  16843
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