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Theorem f0bi 5750
Description: A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
Assertion
Ref Expression
f0bi  |-  ( F : (/) --> X  <->  F  =  (/) )

Proof of Theorem f0bi
StepHypRef Expression
1 ffn 5713 . . 3  |-  ( F : (/) --> X  ->  F  Fn  (/) )
2 fn0 5682 . . 3  |-  ( F  Fn  (/)  <->  F  =  (/) )
31, 2sylib 196 . 2  |-  ( F : (/) --> X  ->  F  =  (/) )
4 f0 5748 . . 3  |-  (/) : (/) --> X
5 feq1 5695 . . 3  |-  ( F  =  (/)  ->  ( F : (/) --> X  <->  (/) : (/) --> X ) )
64, 5mpbiri 233 . 2  |-  ( F  =  (/)  ->  F : (/) --> X )
73, 6impbii 188 1  |-  ( F : (/) --> X  <->  F  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398   (/)c0 3783    Fn wfn 5565   -->wf 5566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-fun 5572  df-fn 5573  df-f 5574
This theorem is referenced by:  f0dom0  5751  map0e  7449  2ffzoeq  32715
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