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Theorem f0dom0 6792
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 6717 . . . 4 (𝑋 = ∅ → (𝐹:𝑋𝑌𝐹:∅⟶𝑌))
2 f0bi 6791 . . . . 5 (𝐹:∅⟶𝑌𝐹 = ∅)
32biimpi 216 . . . 4 (𝐹:∅⟶𝑌𝐹 = ∅)
41, 3biimtrdi 253 . . 3 (𝑋 = ∅ → (𝐹:𝑋𝑌𝐹 = ∅))
54com12 32 . 2 (𝐹:𝑋𝑌 → (𝑋 = ∅ → 𝐹 = ∅))
6 feq1 6716 . . . 4 (𝐹 = ∅ → (𝐹:𝑋𝑌 ↔ ∅:𝑋𝑌))
7 fdm 6745 . . . . 5 (∅:𝑋𝑌 → dom ∅ = 𝑋)
8 dm0 5931 . . . . 5 dom ∅ = ∅
97, 8eqtr3di 2792 . . . 4 (∅:𝑋𝑌𝑋 = ∅)
106, 9biimtrdi 253 . . 3 (𝐹 = ∅ → (𝐹:𝑋𝑌𝑋 = ∅))
1110com12 32 . 2 (𝐹:𝑋𝑌 → (𝐹 = ∅ → 𝑋 = ∅))
125, 11impbid 212 1 (𝐹:𝑋𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  c0 4333  dom cdm 5685  wf 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  pfxn0  14724  elfrlmbasn0  21783  mavmulsolcl  22557  wrdpmtrlast  33113  fdomne0  48759
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