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Theorem f0bi 6792
Description: A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
Assertion
Ref Expression
f0bi (𝐹:∅⟶𝑋𝐹 = ∅)

Proof of Theorem f0bi
StepHypRef Expression
1 ffn 6737 . . 3 (𝐹:∅⟶𝑋𝐹 Fn ∅)
2 fn0 6700 . . 3 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
31, 2sylib 218 . 2 (𝐹:∅⟶𝑋𝐹 = ∅)
4 f0 6790 . . 3 ∅:∅⟶𝑋
5 feq1 6717 . . 3 (𝐹 = ∅ → (𝐹:∅⟶𝑋 ↔ ∅:∅⟶𝑋))
64, 5mpbiri 258 . 2 (𝐹 = ∅ → 𝐹:∅⟶𝑋)
73, 6impbii 209 1 (𝐹:∅⟶𝑋𝐹 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  c0 4339   Fn wfn 6558  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567
This theorem is referenced by:  f0dom0  6793  mapdm0  8881  fset0  8893  0map0sn0  8924  griedg0ssusgr  29297  rgrusgrprc  29622  sticksstones11  42138  2ffzoeq  47277  f102g  48682
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