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Theorem f0bi 6250
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 6207 . . 3 (𝐹:∅⟶𝑋𝐹 Fn ∅)
2 fn0 6173 . . 3 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
31, 2sylib 208 . 2 (𝐹:∅⟶𝑋𝐹 = ∅)
4 f0 6248 . . 3 ∅:∅⟶𝑋
5 feq1 6188 . . 3 (𝐹 = ∅ → (𝐹:∅⟶𝑋 ↔ ∅:∅⟶𝑋))
64, 5mpbiri 248 . 2 (𝐹 = ∅ → 𝐹:∅⟶𝑋)
73, 6impbii 199 1 (𝐹:∅⟶𝑋𝐹 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  c0 4059   Fn wfn 6045  wf 6046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-fun 6052  df-fn 6053  df-f 6054
This theorem is referenced by:  f0dom0  6251  mapdm0  8041  map0e  8064  griedg0ssusgr  26378  rgrusgrprc  26717  mapdm0OLD  39901  2ffzoeq  41867
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