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Theorem f0bi 5705
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 5670 . . 3  |-  ( F : (/) --> X  ->  F  Fn  (/) )
2 fn0 5641 . . 3  |-  ( F  Fn  (/)  <->  F  =  (/) )
31, 2sylib 196 . 2  |-  ( F : (/) --> X  ->  F  =  (/) )
4 f0 5703 . . 3  |-  (/) : (/) --> X
5 feq1 5653 . . 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 1370   (/)c0 3748    Fn wfn 5524   -->wf 5525
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-fun 5531  df-fn 5532  df-f 5533
This theorem is referenced by:  f0dom0  5706  2ffzoeq  30382
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