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Theorem f0 5441
Description: The empty function. (Contributed by NM, 14-Aug-1999.)
Assertion
Ref Expression
f0  |-  (/) : (/) --> A

Proof of Theorem f0
StepHypRef Expression
1 eqid 2296 . . 3  |-  (/)  =  (/)
2 fn0 5379 . . 3  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
31, 2mpbir 200 . 2  |-  (/)  Fn  (/)
4 rn0 4952 . . 3  |-  ran  (/)  =  (/)
5 0ss 3496 . . 3  |-  (/)  C_  A
64, 5eqsstri 3221 . 2  |-  ran  (/)  C_  A
7 df-f 5275 . 2  |-  ( (/) :
(/) --> A  <->  ( (/)  Fn  (/)  /\  ran  (/)  C_  A ) )
83, 6, 7mpbir2an 886 1  |-  (/) : (/) --> A
Colors of variables: wff set class
Syntax hints:    = wceq 1632    C_ wss 3165   (/)c0 3468   ran crn 4706    Fn wfn 5266   -->wf 5267
This theorem is referenced by:  f00  5442  f10  5523  fconstfv  5750  map0g  6823  ac6sfi  7117  oif  7261  wrd0  11434  ram0  13085  gsum0  14473  ga0  14768  0frgp  15104  ptcmpfi  17520  0met  17946  perfdvf  19269  umgra0  23892  vdgr0  23907  0alg  25859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275
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