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Theorem f0bi 6388
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 6341 . . 3 (𝐹:∅⟶𝑋𝐹 Fn ∅)
2 fn0 6306 . . 3 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
31, 2sylib 210 . 2 (𝐹:∅⟶𝑋𝐹 = ∅)
4 f0 6386 . . 3 ∅:∅⟶𝑋
5 feq1 6322 . . 3 (𝐹 = ∅ → (𝐹:∅⟶𝑋 ↔ ∅:∅⟶𝑋))
64, 5mpbiri 250 . 2 (𝐹 = ∅ → 𝐹:∅⟶𝑋)
73, 6impbii 201 1 (𝐹:∅⟶𝑋𝐹 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1508  c0 4172   Fn wfn 6180  wf 6181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-fun 6187  df-fn 6188  df-f 6189
This theorem is referenced by:  f0dom0  6389  mapdm0  8219  griedg0ssusgr  26765  rgrusgrprc  27089  mapdm0OLD  40916  2ffzoeq  42968
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