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Theorem f0dom0 5987
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 5926 . . . 4 (𝑋 = ∅ → (𝐹:𝑋𝑌𝐹:∅⟶𝑌))
2 f0bi 5986 . . . . 5 (𝐹:∅⟶𝑌𝐹 = ∅)
32biimpi 205 . . . 4 (𝐹:∅⟶𝑌𝐹 = ∅)
41, 3syl6bi 242 . . 3 (𝑋 = ∅ → (𝐹:𝑋𝑌𝐹 = ∅))
54com12 32 . 2 (𝐹:𝑋𝑌 → (𝑋 = ∅ → 𝐹 = ∅))
6 feq1 5925 . . . 4 (𝐹 = ∅ → (𝐹:𝑋𝑌 ↔ ∅:𝑋𝑌))
7 dm0 5247 . . . . 5 dom ∅ = ∅
8 fdm 5950 . . . . 5 (∅:𝑋𝑌 → dom ∅ = 𝑋)
97, 8syl5reqr 2659 . . . 4 (∅:𝑋𝑌𝑋 = ∅)
106, 9syl6bi 242 . . 3 (𝐹 = ∅ → (𝐹:𝑋𝑌𝑋 = ∅))
1110com12 32 . 2 (𝐹:𝑋𝑌 → (𝐹 = ∅ → 𝑋 = ∅))
125, 11impbid 201 1 (𝐹:𝑋𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  c0 3874  dom cdm 5028  wf 5786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4579  df-opab 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-fun 5792  df-fn 5793  df-f 5794
This theorem is referenced by:  swrdn0  13231  elfrlmbasn0  19873  mavmulsolcl  20124
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