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Theorem f0bi 6765
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 6708 . . 3 (𝐹:∅⟶𝑋𝐹 Fn ∅)
2 fn0 6672 . . 3 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
31, 2sylib 217 . 2 (𝐹:∅⟶𝑋𝐹 = ∅)
4 f0 6763 . . 3 ∅:∅⟶𝑋
5 feq1 6689 . . 3 (𝐹 = ∅ → (𝐹:∅⟶𝑋 ↔ ∅:∅⟶𝑋))
64, 5mpbiri 258 . 2 (𝐹 = ∅ → 𝐹:∅⟶𝑋)
73, 6impbii 208 1 (𝐹:∅⟶𝑋𝐹 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  c0 4315   Fn wfn 6529  wf 6530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-fun 6536  df-fn 6537  df-f 6538
This theorem is referenced by:  f0dom0  6766  mapdm0  8833  fset0  8845  0map0sn0  8876  griedg0ssusgr  28994  rgrusgrprc  29318  sticksstones11  41469  2ffzoeq  46546  f102g  47730
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