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Theorem f0 5619
Description: The empty function. (Contributed by NM, 14-Aug-1999.)
Assertion
Ref Expression
f0  |-  (/) : (/) --> A

Proof of Theorem f0
StepHypRef Expression
1 eqid 2435 . . 3  |-  (/)  =  (/)
2 fn0 5556 . . 3  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
31, 2mpbir 201 . 2  |-  (/)  Fn  (/)
4 rn0 5119 . . 3  |-  ran  (/)  =  (/)
5 0ss 3648 . . 3  |-  (/)  C_  A
64, 5eqsstri 3370 . 2  |-  ran  (/)  C_  A
7 df-f 5450 . 2  |-  ( (/) :
(/) --> A  <->  ( (/)  Fn  (/)  /\  ran  (/)  C_  A ) )
83, 6, 7mpbir2an 887 1  |-  (/) : (/) --> A
Colors of variables: wff set class
Syntax hints:    = wceq 1652    C_ wss 3312   (/)c0 3620   ran crn 4871    Fn wfn 5441   -->wf 5442
This theorem is referenced by:  f00  5620  f10  5701  fconstfv  5946  map0g  7045  ac6sfi  7343  oif  7491  wrd0  11724  ram0  13382  gsum0  14772  ga0  15067  0frgp  15403  ptcmpfi  17837  0met  18388  perfdvf  19782  uhgra0  21336  umgra0  21352  vdgr0  21663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450
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