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Theorem f0bi 6726
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 6669 . . 3 (𝐹:∅⟶𝑋𝐹 Fn ∅)
2 fn0 6633 . . 3 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
31, 2sylib 217 . 2 (𝐹:∅⟶𝑋𝐹 = ∅)
4 f0 6724 . . 3 ∅:∅⟶𝑋
5 feq1 6650 . . 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 1542  c0 4283   Fn wfn 6492  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  f0dom0  6727  mapdm0  8781  fset0  8793  0map0sn0  8824  griedg0ssusgr  28216  rgrusgrprc  28540  sticksstones11  40567  2ffzoeq  45567  f102g  46925
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