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Theorem f0 5567
Description: The empty function. (Contributed by NM, 14-Aug-1999.)
Assertion
Ref Expression
f0  |-  (/) : (/) --> A

Proof of Theorem f0
StepHypRef Expression
1 eqid 2387 . . 3  |-  (/)  =  (/)
2 fn0 5504 . . 3  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
31, 2mpbir 201 . 2  |-  (/)  Fn  (/)
4 rn0 5067 . . 3  |-  ran  (/)  =  (/)
5 0ss 3599 . . 3  |-  (/)  C_  A
64, 5eqsstri 3321 . 2  |-  ran  (/)  C_  A
7 df-f 5398 . 2  |-  ( (/) :
(/) --> A  <->  ( (/)  Fn  (/)  /\  ran  (/)  C_  A ) )
83, 6, 7mpbir2an 887 1  |-  (/) : (/) --> A
Colors of variables: wff set class
Syntax hints:    = wceq 1649    C_ wss 3263   (/)c0 3571   ran crn 4819    Fn wfn 5389   -->wf 5390
This theorem is referenced by:  f00  5568  f10  5649  fconstfv  5893  map0g  6989  ac6sfi  7287  oif  7432  wrd0  11659  ram0  13317  gsum0  14707  ga0  15002  0frgp  15338  ptcmpfi  17766  0met  18304  perfdvf  19657  uhgra0  21211  umgra0  21227  vdgr0  21519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-fun 5396  df-fn 5397  df-f 5398
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