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Theorem f0 5695
Description: The empty function. (Contributed by NM, 14-Aug-1999.)
Assertion
Ref Expression
f0  |-  (/) : (/) --> A

Proof of Theorem f0
StepHypRef Expression
1 eqid 2452 . . 3  |-  (/)  =  (/)
2 fn0 5633 . . 3  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
31, 2mpbir 209 . 2  |-  (/)  Fn  (/)
4 rn0 5194 . . 3  |-  ran  (/)  =  (/)
5 0ss 3769 . . 3  |-  (/)  C_  A
64, 5eqsstri 3489 . 2  |-  ran  (/)  C_  A
7 df-f 5525 . 2  |-  ( (/) :
(/) --> A  <->  ( (/)  Fn  (/)  /\  ran  (/)  C_  A ) )
83, 6, 7mpbir2an 911 1  |-  (/) : (/) --> A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    C_ wss 3431   (/)c0 3740   ran crn 4944    Fn wfn 5516   -->wf 5517
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-fun 5523  df-fn 5524  df-f 5525
This theorem is referenced by:  f00  5696  f0bi  5697  f10  5775  fconstfv  6044  map0g  7357  ac6sfi  7662  oif  7850  wrd0  12365  0csh0  12543  ram0  14196  gsum0  15624  ga0  15930  0frgp  16392  ptcmpfi  19513  0met  20068  perfdvf  21506  uhgra0  23390  umgra0  23406  vdgr0  23717
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