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Theorem f0dom0 6327
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 6261 . . . 4 (𝑋 = ∅ → (𝐹:𝑋𝑌𝐹:∅⟶𝑌))
2 f0bi 6326 . . . . 5 (𝐹:∅⟶𝑌𝐹 = ∅)
32biimpi 208 . . . 4 (𝐹:∅⟶𝑌𝐹 = ∅)
41, 3syl6bi 245 . . 3 (𝑋 = ∅ → (𝐹:𝑋𝑌𝐹 = ∅))
54com12 32 . 2 (𝐹:𝑋𝑌 → (𝑋 = ∅ → 𝐹 = ∅))
6 feq1 6260 . . . 4 (𝐹 = ∅ → (𝐹:𝑋𝑌 ↔ ∅:𝑋𝑌))
7 dm0 5572 . . . . 5 dom ∅ = ∅
8 fdm 6287 . . . . 5 (∅:𝑋𝑌 → dom ∅ = 𝑋)
97, 8syl5reqr 2877 . . . 4 (∅:𝑋𝑌𝑋 = ∅)
106, 9syl6bi 245 . . 3 (𝐹 = ∅ → (𝐹:𝑋𝑌𝑋 = ∅))
1110com12 32 . 2 (𝐹:𝑋𝑌 → (𝐹 = ∅ → 𝑋 = ∅))
125, 11impbid 204 1 (𝐹:𝑋𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1658  c0 4145  dom cdm 5343  wf 6120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-fun 6126  df-fn 6127  df-f 6128
This theorem is referenced by:  swrdn0OLD  13718  pfxn0  13766  elfrlmbasn0  20470  mavmulsolcl  20726
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