MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f0bi Structured version   Visualization version   GIF version

Theorem f0bi 6774
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 6716 . . 3 (𝐹:∅⟶𝑋𝐹 Fn ∅)
2 fn0 6680 . . 3 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
31, 2sylib 217 . 2 (𝐹:∅⟶𝑋𝐹 = ∅)
4 f0 6772 . . 3 ∅:∅⟶𝑋
5 feq1 6697 . . 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 1534  c0 4318   Fn wfn 6537  wf 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-fun 6544  df-fn 6545  df-f 6546
This theorem is referenced by:  f0dom0  6775  mapdm0  8854  fset0  8866  0map0sn0  8897  griedg0ssusgr  29071  rgrusgrprc  29396  sticksstones11  41622  2ffzoeq  46702  f102g  47898
  Copyright terms: Public domain W3C validator