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Theorem f0dom0 6558
Description: A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
Assertion
Ref Expression
f0dom0 (𝐹:𝑋𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))

Proof of Theorem f0dom0
StepHypRef Expression
1 feq2 6491 . . . 4 (𝑋 = ∅ → (𝐹:𝑋𝑌𝐹:∅⟶𝑌))
2 f0bi 6557 . . . . 5 (𝐹:∅⟶𝑌𝐹 = ∅)
32biimpi 218 . . . 4 (𝐹:∅⟶𝑌𝐹 = ∅)
41, 3syl6bi 255 . . 3 (𝑋 = ∅ → (𝐹:𝑋𝑌𝐹 = ∅))
54com12 32 . 2 (𝐹:𝑋𝑌 → (𝑋 = ∅ → 𝐹 = ∅))
6 feq1 6490 . . . 4 (𝐹 = ∅ → (𝐹:𝑋𝑌 ↔ ∅:𝑋𝑌))
7 dm0 5785 . . . . 5 dom ∅ = ∅
8 fdm 6517 . . . . 5 (∅:𝑋𝑌 → dom ∅ = 𝑋)
97, 8syl5reqr 2871 . . . 4 (∅:𝑋𝑌𝑋 = ∅)
106, 9syl6bi 255 . . 3 (𝐹 = ∅ → (𝐹:𝑋𝑌𝑋 = ∅))
1110com12 32 . 2 (𝐹:𝑋𝑌 → (𝐹 = ∅ → 𝑋 = ∅))
125, 11impbid 214 1 (𝐹:𝑋𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1533  c0 4291  dom cdm 5550  wf 6346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-fun 6352  df-fn 6353  df-f 6354
This theorem is referenced by:  pfxn0  14042  elfrlmbasn0  20901  mavmulsolcl  21154
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