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Theorem f0 5425
Description: The empty function. (Contributed by NM, 14-Aug-1999.)
Assertion
Ref Expression
f0  |-  (/) : (/) --> A

Proof of Theorem f0
StepHypRef Expression
1 eqid 2283 . . 3  |-  (/)  =  (/)
2 fn0 5363 . . 3  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
31, 2mpbir 200 . 2  |-  (/)  Fn  (/)
4 rn0 4936 . . 3  |-  ran  (/)  =  (/)
5 0ss 3483 . . 3  |-  (/)  C_  A
64, 5eqsstri 3208 . 2  |-  ran  (/)  C_  A
7 df-f 5259 . 2  |-  ( (/) :
(/) --> A  <->  ( (/)  Fn  (/)  /\  ran  (/)  C_  A ) )
83, 6, 7mpbir2an 886 1  |-  (/) : (/) --> A
Colors of variables: wff set class
Syntax hints:    = wceq 1623    C_ wss 3152   (/)c0 3455   ran crn 4690    Fn wfn 5250   -->wf 5251
This theorem is referenced by:  f00  5426  f10  5507  fconstfv  5734  map0g  6807  ac6sfi  7101  oif  7245  wrd0  11418  ram0  13069  gsum0  14457  ga0  14752  0frgp  15088  ptcmpfi  17504  0met  17930  perfdvf  19253  umgra0  23877  vdgr0  23892  0alg  25756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259
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